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CONSTRUCTOR 



A HAND-BOOK 

OF 

MACHINE DESIGN 

BY 

f/reuleaux 

Professor at the Royal Technical High School at Berlin, Royal Privy Conncillor, 

Member of the Royal Technical Deputation, 

Corresponding Member of the Institute of Lombardy and of the Swedish Technical Society, 

Foreign Member of the Royal Academy of Sciences of Stockholm, 

Honorary Member of the Technical Societies of Riga and Erfurt, 

of the Technical Society of Frankfurt a M., of the Society of Arts of Geneva, 

of the Flora Society of Cologne, of the American Philosophical Society 

and of the American Society of Mechanical Engineers. 



With Portrait and over 1200 Illustrations. 



AUTHORIZED TRANSLATION 

COMPLETE AND UNABRIDGED 
From the Fourth Enlarged German Edition 

BY 

HENRY HARRISON SUPLEE, B. Sc. 

Member of the American Society of Mechanical Engineers 
Member of the Franklin Institute. 



NEW YORK 



\ h:. h:. sxjflee 



120 Liberty St. 
1899 






Copyright, I890, by John M. Davis. 



Copyright, 1893, by Henry Harrison Suplee. 



Entered at Stationers Hall. 



A. 



Translator's Preface. 




In presenting to the engineering profession of England and America this translation of 
Reuleaux's Constructor, a few prefatory remarks may be permitted. Although the first edition 
of the German work appeared as long ago as i86i,and translations have been made into French, 
Swedish and Russian, no English translation has hitherto been made, notwithstandine the fact 
that repeated editions and enlargements of the original German work have appeared. 

The translation here given, therefore, is the first presentation to English speaking engi- 
neers of a work which during the past thirty years has acquired the highest reputation over all 
Europe, and is so well known to German reading engineers and students in this country that no 
excuse is needed for its present appearance. 

The freedom with which the author has drawn from English and American sources as well 
as from Continental practice gives the work a value not found in other treatises upon machine de- 
sign, while the vast improvement which has been made by the introduction of the kinematic 
analysis and the resulting classification of the details of the subject, cannot fail to appeal to the 
instructor as well as to the practising engineer. 

The translation has been made from the Fourth Enlarged German Edition of 1889, the 
last which has appeared in the original, and is complete and unabridged in every respect. The 
introduction to this edition is especially worthy of note, as it contains the author's summary of the 
principles set forth in his larger work on Theoretical Kinematics,* and the more so as it includes 
a brief glance at the still wider subject included in his work on Applied Kinematics, as yet 
unpublished in Germany, and embodying a mass of manuscript which it is trusted will at no. 
distant day be given to the public. 

The work of translation has been done with the especial sanction and exclusive authoriza- 
tion of Prof. Reuleaux, by whom also the portrait and special introduction to the American 
edition have been furnished. 

The transformation of the notation of the work from the metric system to the English 
values has involved much labor and while it is too much to expect entire freedom from errors, not- 
withstanding the care which has been given to this portion of the work, it is trusted that but few 
errors will be found. It is especially requested that any corrections which may be found neces- 
sary will kindly be sent to the translator for future use. 

HENRY HARRISON SUPLEE. 
Philadelphia, September, 1893. 



* It is to be regretted that Prof. Kennedy's translation of this valuable work is now out of print, and it is hoped that a 
new edition maybe issued. 



Author's Introduction to the American Edition, 



The present translation of the Constructor places my book before a large circle of 
readers who have been practically active and energetic in the development of machine design, 
for no one of the technical professions has been followed by the English-speaking race with 
more activity and success than that of the construction of machinery. I therefore take pleasure 
in prefacing this book with a few words of special introduction. 

During the series of years in which my Constructor has grown from a small beginning 
to a large volume, the practice of machine construction has also been continuously developing, 
so that in every new edition changes and additions have been necessary. Much new matter has 
been added in this edition to the theoretical portion ; first, in the section on Graphical Statics, 
enabling many numerical calculations to be dispensed with, using in their places graphical meth- 
ods ; second, by the introduction of the methods of Kinematics, or the science of controlled 
movements, a science which reduces the apparently inexhaustible complexity of machine forms 
to a few simple and fundamental principles, the command of which may be of extraordinary 
"Value to the engineer. I am still constantly engaged with the subject of Kinematics, especially 
"with its practical applications, but on account of the pressure of other occupations I have not as 
yet been able to carry out my intention of treating this portion of the subject in a separate work, 
•corresponding to my work on Theoretical Kinematics. The work already published on this 
subject I have therefore characterized as an " outline " of a theory of machines." * 

The simplification of the conceptions concerning machines to which these kinematical 
studies led me, was of such importance that I have introduced the kinematical treatment into 
the Constructor in various places, especially in the latter portion of the book. Even where no 
special reference has been made to it, the theory has been followed, although the proof has been 
omitted in order to avoid burdening the non-theoretical reader with details not absolutely neces- 
sary for the practical application. It is in this manner that kinematical axioms have been intro- 
<iuced into Chapter XVIII., where the subject of ratchets is treated. These were formerly 
considered as devices of only minor importance, but the application of kinematical investigation 
reveals the fact that they are of the very greatest importance, occupying a position in machine 
construction superior to that of any other element or combination, and this notwithstanding the 
apparent simplicity and almost insignificant appearance of the original contrivance. A similar 
treatment has been given to the subject of Pressure organs, Chapter XXIII. Hitherto fluids, 
rsuch as water, steam, gas, etc., have been considered as something apart from the machine, not 
l)elonging to it, but rather introduced from the outside. The idea that fluids, broadly considered, 
are but the exact opposites of tension organs, such as ropes, chains, belts, wire cables, etc., is 
wholly contrary to earlier conceptions, and yet it is just this introduction of the kinematic method 
-which has led to an unexpected insight and very great simplification. An illustration of this is 
seen in the manner in which valves for pressure organs are treated as ratchets. In Chapter 
XVIII. ratchets formed of rigid elements only are considered, but the principles there deduced 
are applied in Chapters XXIII. to XXVI. to fluid elements with most satisfactory results. Since 

* F. Reuleaux, The Kinematics of Machinery. Outlines of a Theory of Machines. Translated and edited by Alex. 
3. W. Kennedy, C. E., London. Macmillan & Co., 1S76. 



INTRODUCTION. v 

the kinematic analysis has shown that such devices as pneumatic tubes, canal locks, and the like, 
both ancient and modern, belong to precisely the same class of constrained combinations as 
steam engines and water wheels, the whole subject has been condensed and simplified in a man- 
ner not possible under the earlier conceptions. The value of the kinematic method is evident in 
in Section 333, where fifty different combinations of pressure organs are gathered together under 
a few and simple fundamental principles. Another instance is shown near the end of the book 
in the discussion of what I have called " Fluid valves." From the time of Hero of Alexandria 
down to the present day, these fluid valves have been used in what is now seen to be a continu- 
ous series of applications of a simple kinematical principle. These important simplifications will 
both excuse and justify the wide departure from previous conceptions which characterizes the 
latter part of the volume. 

In regard to the other and principal object of the work, namely, the treatment of the 
practical construction of machine details, this has not been as consistently and fully revised as I 
had intended and desired ; chiefly owing to the long delay in the completion of the last edition- 
In my lectures I have been able to follow the the technical advances which have been made in 
the detailed construction of bearings, levers, cranks, connecting rods, etc., and discuss them 
accordingly, but in the book itself many of these subjects still appear in the older dress. For 
these imperfections the kind indulgence of the reader is requested, and in the next edition an 
earnest endeavor will be made to bring these subjects up to date. 

To Mr. Henry Harrison Suplee, to whom I have given the exclusive right of translation, 
I take this opportunity to express my particular appreciation of the great care and extraordinary 
accuracy which he has displayed in the production of this English version, and also my gratifica- 
tion at the care which has been given to the printing and the reproduction of the illustrations. 
Mr. Suplee has recalculated and transformed all the formulee and numerous tables into the 
English system of measurements, and also reworked all the examples, and has shown in this 
portion of the work a patience that deserves especial recognition. It is a matter of regret that 
the time has not yet arrived for the general acceptance of the metric system in England and 
America, and until such time comes tedious transformations of this sort will often be necessary 
and will merit our gratitude. 

I can only add that it is my earnest desire that the friendly acceptance of my book by 
English speaking engineers may correspond to the magnitude of the labor which has been 
expended in the preparation of this translation, 

F. REULEAUX, 
Honorary Member, American Society of Mechanical Ens^ineers, 
Berlin, February, 1803. 




/^ RECEIVED '', 

L MAY 131908 ^) 



Introduction to the Fourth German Edition. 



The fourth enlarged edition of the Constructor is 
presented to its readers much later than I had hoped. 
As some excuse for the delay I plead the great labor 
involved in the re-arrangement of more than half the 
book. As already explained, it has been my intention 
to re-arrange the matter upon a kinematical basis. 
It was not, however, entirely due to this re-arrange- 
ment that the work was delayed, but also to the fact 
that nearly one-half of the work had to be re-written. 
In many places I found almost everything lacking to 
make what I had previously determined upon, namely, 
a complete and consistent whole, and much more was 
needed than I had imagined. In addition to these 
shortcomings the spirit of invention has been more ac- 
tive than ever during the past few years and advanced 
at such a rapid rate that I could by no means overtake 
it. It is hoped that these conditions may be accepted 
as at least a partial excuse for the delay and for the 
shortcomings of the work. 

The first point to which I desire to call attention in 
the new matter is the subject of Ratchets, which upon 
closer examination will be found to be the most im- 
portant of all forms of driving mechanism. This sub- 
ject has not until now been treated as an element of 
construction, it having been apparently overlooked that 
those forms of driving mechanism in vi'hich pawls and 
ratchet wheels form a part, are in reality a most 
important and prolific class. Special forms have in- 
deed been treated mainly as checking devices but 
without any attempt to indicate the general principles, 
or wide extent of the construction. Locks, in spite of 
their universal use and of the high order of inventive 
talent devoted to them, have had no analytical treat- 
ment, but have been relegated to the domain of tech- 
nology rather by accident than otherwise, and from 
Prechtl to Karmarsch and his followers, have been giv- 
en an intelligent but by no means fundamental treat- 
ment. Gun locks, although having a similar name to 
door locks, have a very different construction, but have 
found no resting place in technical literature. It has 
often been observed that while we place in the hands 
of our soldiers the modern rifles and cannon, there is 
no place in the head for them, either in machine shop 
training, in machine design, in applied mechanics or 
in technology, or indeed anywhere. In §252 I have 
placed them in that class which I have termed Locking 
Ratchets where they fall into their proper place as 
members of the great division of ratchet gearing. 
The safety devices for elevators and hoisting machines, 
— Checking Ratchets, I have termed them — have been 



entirely overlooked; books have been written about 

■ them, catalogues and price lists issued, but the funda- 
mental principle of their construction quite overlooked. 
As for escapements of clocks and watches, these have 
been sent hither and thither, now in mechanical text 
books, now in kinematics, now in applied mechanics, 
again in encycloposdias, where their fundamental prin- 
ciple has been entirely lost, their intimate relation to 
ratchet mechanism being hardly noticed. They will 
here be found classified in their proper place in §258. 
Many of the readers of previous editions may shake 
their heads at this statement, but an examination of 
the fourth edition will show how the action of the pis- 
ton engine is similar in principle to a watch escape- 
ment, the action of the slide valve being practically 
identical with the anchor of the escapement, see §§324, 
325, pp. 228-232. It has only been by more recent iri.- 
vestigations that I have become convinced of the 
relations of these various forms of escapements. The 
correctness of this position will be confirmed by 
comparing the the pneumatic postal tubes, canal locks, 
sluices, hydraulic cranes and numerous other hydraulic 
devices, hydraulic riveting machines, and all the many 
kinds of direct-acting steam pumps ; these and many 
others, when considered from the present point of view, 
arrange themselves in a complete and orderly manner 
as true escapements. The similarity is especially well 
marked in the case of a deep mine pump, of which 
the successive puffs of the exhaust are not infrequently 
used by neighboring dwellers to indicate intervals of 
time ; the steam end practically as well as theoretically 
becoming a time-piece. Nay, more : I am convinced 
that it is not a pure accident that throughout the cen- 
turies in which the delicate clock escapement has been 
known, the steam engine has so slowly developed ; 
for although both the clock and the engine are in prin- 
ciple escapements, yet in the clock there is an escape- 
ment of precision, and in the steam engine an 
escapement of force,* but both devices are theoretically 
a solution of the same problem. Closely allied to the 
steam engine are the various water pressure engines, 
and water pumps, which as I have shown in § 319, are 
truly continuous ratchet trains. From the ratchet to 
the escapement, however, what a long, long gap ! 
The water pump and hydraulic pressure engine differ 
from each other only in the different motion and 
action of the valves — and yet the inventive genius of 



*In my Theoretical Kinematics, I have considered the steam eng;ine as a 
reciprocating running; ratchet train, but I have since perceived this classifica- 
tion to be incorrect, and therefore desire to emphasize its proper classification 
here. 



INTRODUCTION. 



mankind required over two thousand years to make 
that little step, (see § 325). How important, then, to 
make this fundamental connection clear I 

Another important, and hitherto neglected subject, 
is that of the more recent steering- devices, which move 
in either direction, or remain at rest, as required. This 
principle has found many applications in power steer- 
ing gear for vessels, and has even made possible the 
solution of the difficult problem of guiding the auto- 
matic fish-torpedo at a determinate depth. It is not 
surprising that uncertainty should exist as to the theo- 
retical classitication of these devices. I have, how- 
ever, shown that they are properly considered as 
escapements, and, in fact, as escapements of a special 
kind, which I have termed "adjustable" escapements. 
Such adjustable escapements of rigid construction are 
shown in § 259, and those constructed with pistons 
and fluids, in § 329. 

The chapter upon Ratchet Gearing is not only en- 
tirely new, but it has also involved a new and more 
elaborate treatment of many subjects discussed in 
earlier chapters of the book. These I here only name : 
Screw thread systems in Chapter IV. ; Thrust-bearings 
for screw propeller shafts ; Columns ; Long distance 
shafting transmission, etc., in § 351 ; Couplings, Fric- 
tion gearing ; Transmission of motion by toothed gear- 
ing (p. 128); Spiral gears (p. 141): Globoid gearing 
(p. 142), Proportions of gearing, (§ 226-§ 228). Rat- 
chet wheels are treated in a similar manner to spur 
gear wheels, to which they bear a close relation, 
(§ 246). 

From this point the book takes a fresh start, with 
the discussion of another species of machine elements, 
namely. Tension organs, as I have termed them, 
(Chapters XIX. to XXII). While the elements pre- 
viously considered approximate so closely to rigidity 
that they may properly be termed rigid elements, those 
which follow possess the peculiarity that they are only 
adapted to resist tension : these elements include 
cords, ropes, wire, bands, belts, chains, etc. In § 262 
it is shown how these are used in connection with 
other elements in three distinct ways, as for "guiding," 
for "winding," and for "driving." An examination 
of pages 182 to 176 will make the importance of this 
subject evident, and shows its scope to be far greater 
than might at first have been expected. The important 
distinction betvifeen the functions of "driving" and 
"guiding" is shown in the disct'ssion of the differen- 
tial tackle and the ordinary system in connection with 
Fig 813, (p. 176). 

In discussing Cord Friction (§ 264) I have attempt- 
ed to show by a graphical representation relations not 
otherwise easy to make clear. In § 268 I have called 
attention to some points which should be considered 
in connection with stiffness of ropes. The subject of 
pocketed sheaves has been treated in connection with 
chains, and also the chain system of boat propulsion. 



In the chapter upon Belt Transmission, is intro- 
duced a new subject and one which appears to me of 
great importance, and which I have called "Specific 
Capacity." By its use it is possible to facilitate very 
greatly the calculations of Belting, Rope Transmission, 
Water Transmission and even Shafting, and bring them 
to a comparable basis, (see § 349 and § 351). 

The discussions of Hemp and Cotton Rope trans- 
mission are both new, and that of Wire Rope greatly 
enlarged over previous editions. By the introduction 
of the subject of the "mean deflection" (p. 198) and 
the diagram (Fig. 884) the question of the deflection is 
greatly simplified, and a graphical solution is also 
given. Transmission with inclined cables, which in 
previous editions was only given an approximate 
solution, unsuited for long spans, is here accurately 
discussed (assuming the catenary as a parabola) and 
extended to long stretches of cable. This has been 
done in view of the use of rope transmissions and 
telegraph cables over valleys, etc. 

Next follows my system of " Ring Transmission " 
by wire rope. This offers great advantages over the 
previous system of line transmission, and has met 
with much success in Germany, Austria, and Switzer- 
land, as well as in America ; and further discussion of 
it will be given hereafter. The use of chain transmis- 
sion in mines, both in Germany and elsewhere, is dis- 
cussed. The subject of brakes brings the book to 
another point where a fresh start must be made. 

The third group of machine elements includes 
those called "Pressure Organs," and those are treated 
in Chapters XXIII. to XXVI. These are directly opposed 
to tension organs, since they are only capable of resist- 
ing compression, and include not only fluids, both 
liquid and gaseous, but also granular materials, etc. 
(§308). 

Although these elements have been primarily ar- 
ranged in a manner adapted for a practical hand book, 
I believe that my theoretical treatment of the subject 
will also find acceptance, and hence have here included 
the essentials of the theory also (see § 319). Pressure 
organs are serviceable not only in machines, but also 
for the transmission of force and motion ; by them we 
can control the motion of a force in a determinate 
path and with a determinate velocity quite as well as 
with rigid elements, and indeed upon closer inspection 
we perceive that pressure organs are used in nearly all 
the most important prime movers, (steam engines and 
hydraulic motors), and hence they are surely entitled 
to be classed among machine elements. The extent 
to which this conception facilitates the subject of ma- 
chine construction will be seen by an examination of 
the latter part of this volume. 

I have thought it advisable to give also at this time 
a general review of the result of my labors in the 
field of Kinematics. These have been fully and thor- 
oughly given in my lectures for the past twenty-five 
years, and are therefore not new to my immediate 



VJU 



INTRODUCTION. 



pupils, while the publication of my Theoretical Kine- 
matics has placed the the theory before a larger circle 
of scientific readers. I cannot assume, however, that 
the readers of this practical hand book are all familiar 
with the above mentioned work, and I therefore give 
the following abstract, covering the most important 
portions of my treatment of the subject. 

;)! si: :(: * * * 

Motion and the effects which are dependent upon 
motion form the subject of the study of Scientific Me- 
chanics ; and hence to it belongs properly the problems 
of motion in machinery. The motions in a machine, 
however, may be distinguished from others in that 
they can be treated independently of the material 
parts of the machine, and of the forces acting upon 
them. The important bearing which this separation 
gives to the subject of machine construction was per- 
ceived about one hundred years ago, but has made 
small progress during the century and has only re- 
cently been taken up [10-23].* 

I took up this subject in 1862, laying down the 
principles in my lectures; in 1864 first propounded 
them publicly before the convention of the Swiss 
" Na/ur/orscher" a.n.6. their German guests; first pub- 
lished them serially in the Berli7ier ]'erhandlungen in 
1865, and finally in 1872-75 published my book en- 
titled ' ' Theoretische Kineinaiik. " 

The modern discussion of these principles begins 
with the publication, by the celebrated physicist Am- 
pere, in 1830, of his Essat sur la Philosophie des Sci- 
ences, in which he gave the subject the distinctive 
name Kinematics (Cinematkque) , which name is well 
derived from the Greek kmeo, to drive, to constrain, 
since it treats of constrained or controlled motions. 

I have defined the term Kineniaiics [40] as "the 
study of those arrangements of the machine by which 
the mutual motions of its parts, considered as changes 
of position are determined." This I have divided into 
to parts: ''Theoretical" and "Applied" Kinematics, 
the former treating of the general and fundamental 
principles, and the latter of their practical applications. 

a. Theoretical Kinematics. 

It is this branch of the subject which is treated in 
my well-known book " The Kinematics of Machinery.'' 
The following is a condensed analysis of the treatment 
there expanded at greater length : 

1. A material system having motion within itself 
I call a machinal system, as may be determined ac- 
cording as the motion is constrained or not [32]. 

2. Motions can only be constrained by forces. 
These forces differ in the two systems, since in the pure 
machinal system sensible and latent forces enter into 
equilibrium with each other, while in the pure cosmi- 
cal system sensible forces enter into equilibrium with 
sensible forces, [33]. It therefore follows that the 



two systems can not always be accurately determined 
[34].* The terms "latent" and "sensible " are here 
used in a similar sense as in thermal physics. Latent 
forces are those which exert internal resistance to de- 
formation of a body under the action of external 
forces ; sensible forces are those which act upon the 
body from without [},l\. 

3. The motions of the machine can be logically 
controlled according to a predetermined conception, 
since the action .of all external forces which do not 
tend to produce the desired end can be opposed and 
neutralized by latent forces [35]. 

4. From the preceding follows the definition of a 
machine : — 

A machine is a combination 0/ resisient bodies so 
arranged that by their means the mechanical yorces of 
nature can be compelled to do work accompanied by cer- 
tain determinate motions [35, 50, 203]. 

5. If we consider the machine to be made of rigid 
■materials and neglect its mass, we need only take into 
account geometrical considerations [42]. If a body 
A, by means of latent forces, is to be prevented from 
being put in motion by any external forces (case 3), it 
must be held in a stationary position by at least one 
body B. The body B, then acts as the envelope of A, 
and conversely A is the envelope of B, the relation 
being a reciprocal one. There are also reciprocal 
envelope forms possible between the bodies A and B 
tor a relative motion, which shall exclude all other 
relative motions [43]. Such a pair of bodies, I have 
called a kinematic pair of elements and a machine 
consists solely of bodies which thus correspond, pair- 
wise, reciprocally [43 J. 

6. In order to obtain a determinate motion in a 
given space by means of a kinematic pair of elements, 
one of the elements of the pair must be held at rest 
with regard to the given portion of space under con- 
sideration. The relative motion of the moving ele- 
ment to the fixed one will then be that of absolute 
motion, so far as the given portion of space is con- 
cerned [43]. 

7. The choice between the two elements as to 
which shall be stationary and which movable is not 
limited ; the substitution of the fixed for the moving 
element I have called the inversion of the pair [93]. 

8. The control which can be exercised over a de- 
terminate motion in this manner is not mathematically 
exact but only approximate (case 5) because the latent 
forces of bodies can only be brought into action by 
their deformation. If however, the elements are 
made of materials which possess a high degree of 
resistance and are given proper dimensions (machine 
construction) the deformation can be kept within such 
small limits as to be practically insignificant, and the 
result considered as determinate [33]. (Compare cases 
46 to 49, below). 



* The numbers in brackets refer to the pages of Reuleaux's " Kinematics 
of Machinery. Translated by Prof. A. B. W. Kennedy. London, Macmillan 
& Co., 1876. 



* Tiie internal forces of a moving system form the subject of d' Alem- 
bert s principle. 



IXTRODUCTION. 



IX' 



9. Each element of a kinematic pair may be rig- 
idly combined with an element of another similar pair 
without interfering with the relative motion of the 
separate pairs. In this manner a large number of pairs 
of elements may be arranged in a series, so that each 
element of a pair is firmly connected with an element 
of another pair. Such a series of pairs of elements re- 
turns upon itself, resembling a chain [46], consisting 
of links connected together. I have called such a 
series a kinematic chain, and the body which is formed 
by the junction of the elements of two different pairs 
is a link of the kinematic chain [46]. There are 
therefore as many links as there are pairs of kinematic 
elements. 

ID. A kinematic chain may close or return upon 
itself in various ways ; among these is one in which 
every alteration in the position of a link relatively to 
the one next to it is accompanied by an alteration in 
the position of every other link relatively to the first 
[46]. In such a chain each link has only a single 
relative motion with regard to every other link. Such 
a kinematic chain I call a constrained closed — or sim- 
ply a closed chain [46]. 

11. A constrained closed kinematic chain compels 
a definite determinate motion in a given portion of 
space when one link of the chain is fixed, with regard 
to this given portion of space. A closed kinematic 
chain of which one link is thus made stationary, is 
called a mechanism [47]. 

12. A constrained closed kinematic chain, there- 
fore, can be formed into a mechanism in as many 
ways as it has links [47]. The substitution of the 
stationary link of a kinematic chain for another link I 
have called its inversion. 

13. A kinematic chain may have so few members 
and be closed in such a manner that the links can have 
no motion relative to each other, and that the pairs 
themselves do not have their own motion. This I 
have termed fixed closure [485]. 

14. The manner of closure of the chain can be 
chosen so that adjoining links can have more than one 
relative motion. This I have called unconstrained 
closure [485]. 

15. A kinematic chain in which a series of pairs of 
elements are arranged in the stated manner, but of 
which the first and last elements are not connected, I 
have called an open chain. 

16. Kinematic chains of the kinds above men- 
tioned can be combined with each other, forming con- 
structions which may be called compound chains. 
These may have constrained, unconstrained, or fixed 
closure or may be open chains. The same conditions 
exist for these as for the previously described chains, 
which may for sake of distinction, be called simple 
chains. 

17. From the preceding we may give the follow- 
ing general definition of a mechanism, as follows : 



A mechanism is a closed kinematic chain of which 
one link is fixed : this chain is compound or simple, 
and consists of kinematic pairs of elements ; these 
carry the envelopes required for the motion which the 
bodies in contact must have, and by these all motions, 
other than those desired in the mechanism are pre- 
vented [50]. 

18. From all that has preceded, it is apparent that 
the investigation of the motions in machinery is a 
subject which is based in great part upon geometry. 
This has been treated as a separate subject of Phoron- 
omy, or the study of geometrical motion. The most 
important principles of this subject I have treated in 
Chapter II. of my "Theoretical Kinematics," with 
applications to constrained as well as cosmical mo- 
tions [56 to 85]. It is there shown that all relative 
motion can be considered as that of a pair of ruled 
surfaces, so that the motion is reduced to a rolling of 
the two ruled surfaces upon each other, and under cer- 
tain circumstances with a simultaneous endlong slid- 
ing upon each other of the generators which are in 
contact. These rolling surfaces, for which previously 
no special name had been used, I have called axoids, 
the combined sliding and rolling motion being termed 
twisting. When rolling motion is absent only sliding 
remains, when on the contrary, the sliding is omitted 
only the motion of rolling remains. In the latter case 
certain sections through the axoids give curves which 
twist upon each other, or roll with a cross sliding 
action. The combined points of these curves form 
centres of rotation or poles about which, as instanta- 
neous centres, both bodies turn. These centres or 
poles travel in the paths of the aforesaid curves 
whence the latter may be called pole-paths (Polbahnen) 
or centroids. * The study of axoids and centroids will 
greatly extend the range of phoronomic researches. 

19. In order to pass from the general principles to 
the special applications of kinematics, further consid- 
eration must be given to the elementary pairs, The 
simplest form must necessarily be that in which the 
corresponding envelopes actually surround, one the 
other, and such I have called a closed pair. Of this 
there are but three forms : i, the twisting pair (screw 
and nut); 2, the turning pair (pin and collar); 3, the 
sliding pair (full and open prism, or prism pair [91]- 
The two latter may be considered as particular cases- 
of the first. In all three no change in the character of 
the motion is caused by inversion (case 7). 

20. In a pair of elements it is not always neces- 
sary to use all of both envelope forms. The question, 
of the minimum number of points necessary to insure 
resistence to disturning forces, I have discussed in § 17 
of my Kinematics, under the title: "The Necessary 
and Sufficient Restraint of Elements." 



* The term "centroid," due to Prof. Clifford, and used by Prof. Kennedj 
in bis translation of the " Kinematics," will be hereafter used as the transla- 
tion of Polbahn.— Trans. 



INTRODUCTION. 



21. We have thus far omitted from consideration 
such elementary pairs as are not closed. These pos- 
sess the general property of giving a change in the 
■character of the motion when they are inverted. I 
have called them "higher" pairs of elements [115], 
and conversely the closed pairs may be termed "low- 
er " pairs. It is only in special cases that no change 
occurs in the character of the motion by inversion of 
higher pairs. , A series of higher pairs, for the most 
part entirely new, has been discussed in § 21 of my 
Kinematics. 

22. I have given (§ 30 to § 39) seven geometric 
methods of determining the restraining bodies for 
higher pairs, many of which were already known, 
but which were then for the first time grouped into one 
g-eneral system. 

23. Incomplete pairs [169] are those which are 
not entirely closed by the latent forces, but are partly 
closed in some other manner. Examples of these are 
half-journal bearings, in which the weight of the parts 
is used to keep the journal down in its bearing ; knife- 
bearings for scale beams, the V bearings for the beds 
of planing machines, etc. Pairs may also be closed 
by the action of springs or other external forces. The 
closure of a pair of elements in this manne-r I have 
termed "force closure." This form of closure can 
■only be used when the disturbing forces are not sufli- 
■ciently great to overcome the closing force. 

24. Force closure also finds application in higher 
pairs of elements. An important example is found in 
the driving wheels of locomotive engines, and another 
still more important, in the axoid rolling action of 
friction wheels. (.See Chapter XVI. of this volume.) 

25. The application of force closure can be car- 
ried still further. By its application we are enabled to 
utilize two classes of elements which are only capa- 
ble of opposing resistance in one direction (case S). 
These are what I have called "tension organs" and 
"pressure organs," (see § 261 and § 308 of this vol- 
ume). These I have grouped together as " flectional '' 
kinematic elements [173]. They include a long series 
of most useful machines, such as belt and rope trans- 
mission systems, pumps, water-wheels, etc., all in- 
volving the principles of force closure. 

26. Force closure may be used in a dynamical 
as well as in a statical manner, as in the case of an 
engine crank which is carried over the dead centre by 
the action of the fly wheel [186]. 

27. In such cases the closure may also be effected 
by means of another kinematic chain used in combi- 
nation with the first [178]. This I have called chain 
closure. An example is found in a double engine with 
cranks at right angles. 

28. The preferable form of chain-closure is that in 
which similar elements are employed. This occurs 
(case 25) when one force-closing chain is used in con- 
nection with another of the same kind, the two being 
so combined that each supplies the necessary closing 



force for the other ; whence it follows that the sensible 
and latent forces in the two chains counteract each 
other in the same manner as if they were composed of 
rigid elements. [§44, "Complete Kinematic Closure 
of the Flectional Elements."] Examples of this are 
found in the ordinary belt transmission, and in the so- 
called "water rod." By means of this method of 
closure, which is destined to be much more widely 
us«d than heretofore, the applications of flectional 
elements have been greatly extended for purposes to 
which rigid elements are not adapted, such as the 
transmission of force in a path of constantly changing 
direction, as in the tise of high pressure water trans- 
mission systems through pipe conductors. 

29. Finally a kinematic chain may be closed by 
the application of springs [176]. These may be so 
constructed as to oppose resistance in a number of 
chosen directions, but not in all directions ; e. g. , both 
tension and compression, also bending in one plane, but 
not in a second plane at right angles to the first. This 
latter condition is seen in the case of flat or plate 
springs, also in the plate link shown in Fig. 507 of this 
book, where the spring acts as a substitute for a pin 
connection. In the plate link the force closure and 
complete kinematic closure are replaced by chain 
closure. Another example is found in the Emery 
Scale, Fig. 789c. 

30. The pairing of flectional with rigid elements 
may be assumed, a priori, to be practicable in the same 
manner as that of rigid elements [544]- 

31. If the principles of investigation, however 
they ma.y be set forth, are correctly based, they should 
when applied to the historical development of ma- 
chines, shed a light upon the whole subject from the 
rude attempt at invention to the highest attainments of 
mechanical ingenuity. This subject I have discussed 
as a "Sketch of the History of Machine Develop- 
ment, [201 to 246], in which the substitution of pair 
closure for force closure is made most apparent. 

32. In order to facilitate the elucidation of the 
action of machinery, and to abridge the labor of 
the application of the preceding methods, it became 
necessary to devise a system of kinematic notation. 
This is given in Chapter VII., pages 247-273, of 
my Kinematics of Machinery. The elements are 
designated by capital letters, of which twelve are 
required, and the relations of these are indicated by 
auxiliary symbols derived for the most part from 
those already used in mathematical notation. For 
the symbolical representation of the kinematic chain 
I have also introduced the conception of an order 
in which each pair in the chain is numbered from 
I upwards, and the links represented by the small let- 
ters from a onwards [270-273]. The pair and the link 
at which the numbering and lettering is to begin may 
be agreed iipon previously, as well as the direction in 
which they are to proceed. The link between the 
pairs I and 2 will then be indicated by a ; that between 



INTR OD UCTION. 



XI 



2 and 3, by h, etc. For instance, the connecting rod 
and crank device, shown in Fig. 1022 of this book, is 

a 

indicated by the formula (C^ " P-"-) ~. Translated, this 
means that the kinematic chain of the mechanism con- 
sists of three parallel, closed cylinder pairs, and one 
closed prism pair at right angles to them ; that it con- 
tains four links, which I have called the crank, the 
coupler, the slide and the link, and designated by a, 
b, c, and d ; that this chain is converted into a mech- 
anism by the link d being held fast ; that the right line 
from the centre of the bearing 2 to the end of the 
coupler b (the connecting rod) moves around the axis 
I of the crank shaft, and that the crank a is driven, by 
means of the coupler b, by the slide (cross-head) c. 
This is certainly expressing very much by means of 
very few symbols, dispensing with long and compre- 
hensive definition. According to case 12, this chain 
can be converted into three other kinds of machines, 
symboHcally indicated by : (C^" PJ-)^ (C3" P-L)^ etc. 
These symbols have as yet been used but little by 
practical designers, but those who have made use of 
them have found them brief and accurate both for 
writing and for descriotion otherwise requiring much 
longer explanation. 

33. The application of the system of symbols 
leads to vsrhat I have termed "Kinematic Analysis," 
[Chapter VIII. ] The application of this analysis to 
the so-called "mechanical powers," [275-283] leads to 
interesting conclusions, this is also the case with the 
cylindric crank chain [283-341], which taken in con- 
nection with Chapter V., yields a wealth of valuable 
results. 

34. This is followed by an analysis of "chamber- 
crank " trains, Chapter IX. In this, it is shown that 
upwards of a hundred pressure organ machines, 
hitherto considered as separate inventions, have a sys- 
tematic relationship dependent largely upon kinematic 
inversion ; and a number of difficulties are cleared up. 

35. In Chapter X. the subject of the so-called 
" chamber-wheel " trains is analyzed ; the principles of 
which 1 had previously investigated in 1868. 

36. Finally, in Chapter XL, is given the Analysis 
of the Constructive Elements of Machinery, including 
a brief investigation of ratchet mechanisms. At the 
time this portion was written my investigation of that 
subject, however, had not been carried to any great 
extent, and in the present volume for the first time have 
I set forth the extraordinary and varied importance of 
ratchet mechanism. 

37. To this subject is added an analysis of the 
complete machine [486-526], in which the strict limits 
of theoretical kinematics are frequently overstepped and 
encroachments made upon the domain of applied kine- 
matics. The older ideas of the "receptor" the "com- 
municator," and the " tool " are examined and rejected 
and machines classified as "place-changing" and 
"form-changing" machines. This classification will 



be found to possess a decided value and will be referred 
to again. (Cases 42 to 4y.) 

l?>. Kinematic Analysis has as a necessary coun- 
terpart Kinematic Synthesis. This has been already 
seen (cases 19, 21, 30) in the application of pairs, 
chains and mechanisms to given machinal purposes. 
Kinematic synthesis may also be called a theory of the 
invention of mechanisms. This it can only be, how- 
ever, in a limited sense. It can in no case enable the 
genius of the inventor to be dispensed with, but by the 
aid of this theory his scope can be greatly extended. 
The application of synthesis to problems which have 
already been solved may also point the way to the so- 
lution of others as yet undetermined. 

In discussing this synthesis, I have grouped the 
pairs of kinematic elements into 21 orders [538-544] by 
means of which the determination of the greater num- 
ber of kinematic chains and dependent mechanisms 
may be made ; also eight classes of simple chains. 
The application of synthesis may be made in two 
forms, the direct and the indirect, and these again into 
general and special synthesis. Of these the indirect 
synthesis is the most useful [529]. It is my expecta- 
tion that this theoretical exposition of the subject, 
which I cannot expect to extend further, but by means 
of which I have been able to devise a number of new 
mechanisms, may find many successful applications 
by others. 

b. Applied Kinematics. 

39. Applied Kinematics is not so much to be con- 
sidered as standing in opposition to theoretical kine- 
matics as it is included in it. In fact, applied kinematics 
has existed as a study for a long time, as in the treatise 
of Monge, without the existence of any theoretic 
foundation. That such a treatment of kinematics may 
be very useful for a time is readily admissible, but an 
ex post facto theoretical discussion may seem of little 
value to the practical man. Indeed my highly es- 
teemed former preceptor, Redtenbacher, considered an 
actual theoretical treatment of the movements of ma- 
chinery to be an impossibility. 

Under these circumstances I did not feel inclined 
to follow the ' ' Theoretical Kinematics " hastily with a 
treatise on the applied science. For this purpose it 
was not possible to arrange all the various forms of 
machines under the new classification hurriedly and 
properly in permanent form. Notwithstanding the 
simplicity of the preceding system, its application de- 
veloped many difficulties and required a succession of 
researches with which even my immediate pupils are 
not fully acquainted. A not inexcusable impatience 
on their part has led me to have my investigations in 
applied kinematics multiplied for a limited circulation 
although the matter was incomplete. I gave this per- 
mission reluctantly and with the condition that only 
a limited number of impressions, to be considered as 
" manuscript, " should be circulated. In this manner 



INTRODUCTION. 



four parts of the work have appeared, the last consist- 
ing entirely of the application of the symbols to lecture 
room models. The result of such premature publica- 
tion cannot always be foreseen by those who have 
urged it, but for the misunderstandings which have 
arisen from this source I can only express my regret. 
In the meantime I have since 1882 been engaged 
in the partial application of the principles of kinemat- 
ics to this book in such a manner as to avoid burden- 
ing the reader with theoretical matter, which would be 
contrary to the purpose of the work. The most im- 
portant subjects to which the kinematic method has 
been applied are here briefly noted. 

40. With the great extension of modern mechani- 
cal engineering we find that the various mechanisms, 
(the number of which as we have seen is not great), 
are given a great variety of applications. It is the 
object of applied kinematics to furnish a clear distinc- 
tion between the various methods of practical applica- 
tion. It is apparent that the preceding analysis does 
not extend to this point, since it does not include the 
subject of the method of constraint, but only treats of 
the combination of the elementary parts which are 
involved. We may therefore properly term it the 
Elementary Kinematic Analysis. As a counterpart for 
this in applied kinematics we may place the subject of 
another analysis which relates to the conditions of 
motion in a given train, and which may be called 
Train Analysis, or the Analysis of Trains. This anal- 
ysis is not intended to solve anev^^ the construction of 
the various trains, but rather to elucidate clearly their 
method of action ; a train consisting of a closed group 
of elements and bearing the same relation to a machine 
as an atom does to a molecule. 

41. Train analysis does not admit of an arrange- 
ment logically similar to the elementary analysis, but 
possesses a new and different order. This is due to the 
fact that the elements of which trains are composed 
occur only in pairs, while the trains of which machines 
are composed are considered singly. In Vose's pump, 
for example. Fig. 979*, there are two ratchet trains 
combined in one machine, vv^hile in Downton's pump, 
Fig. 979^ there are three trains. 

42. The various methods of tain action may be 
divided into four principal kinematic divisions, viz. : 
Guiding, Storing, Driving, and Forming, §333.* The 
first three divisions are " Place-changing " and the last 
is "Form-changing." 

43. Various forms of guiding devices may be 
mentioned ; linkages by means of which curved paths 
are obtained, parallel and straight line motions, also 
"position motions," as I have termed those by means 
of which a system of points may be transferred to 
another position parallel to the first. Guiding devices 
can be constructed from kinematic chains of every 



* 5" § 333, tl^s second of these has been translated "Supporting," and the 
English language lacks a suitable equivalent for '' Haltung" but in a corres- 
pondence with the author, the above has been adopted Trans. 



kind. It was by means of examples with chains for 
this purpose that the general conditions of motion in 
theoretical kinematics were illustrated, and the same 
conditions belong also to applied kinematics. 

34- Storing includes those especial machine organs 
by means of which work can be accumulated and the 
supply drawn upon for later use. This, until now has 
not been considered as a special mechanical concep- 
tion, although it has had numerous applications. Stor- 
age of power may be accomplished in three quite 
different ways. 

a. By means of rigid elements, this being statical 
or dynamical. Examples of statical storage are found 
in elevated weights, compressed springs, etc., and of 
dynamical storage in fly wheels, or pendulums. One 
of the oldest forms of dynamical storage is the old- 
fashioned spindle [216]. 

b. By means of tension organs, acting by winding 
the tension organ upon a drum or pulley. Examples 
are seen in tower clocks, etc. 

c. By means of pressure organs. These are the 
most frequently used, and examples include tanks for 
water, oil, gas, air, stearti, also hydraulic accumulators. 

45- Driving. In this term I include the transmis- 
sion of motion within a single train and also from one 
system to another. As "guiding " includes the control 
of the path of a point, "driving" considers the control 
of the velocities of various points in their paths. Ex- 
amples in this branch of applied kinematics are those 
which take into consideration the velocity of the var- 
ious parts of a mechanism. (See the close of Case 38). 

46. Forming, includes the working of materials by 
means of machine tools. This fourth division is the 
richest of all, and offers the widest range to the genius 
of invention. This operation takes place by the action 
of the tool upon the material, or as I have called it, 
the '' work piece " [495]- In form changing machines, 
the work-piece is a part or the whole of a kinematic 
link, and is paired or chained with the tool by so ar- 
ranging the latter that it itself changes the original 
form of the work-piece into that of the envelope cor- 
responding to the motion in the pair or linkage em- 
ployed [495]. We can distinguish between three forms 
in which this action can occur. 

a. The tool is hard and operates by cutting the 
material from the work-piece which lies without the 
envelope of the desired form. Examples are found in 
lathes, planers, grinding machines, etc. 

h. The tool is of high resistance so as to be able 
to maintain its form, but does not act by cutting, but 
by pressure upon the yielding work-piece. It follows 
that the material which lies outside of the desired form 
is forced into another part of the work-piece without 
being removed from it. Examples are found in coin- 
ing presses, rolling mills, wire drawing benches, etc. 

c. The tool and the work-piece are both alike 
yielding, and act each upon the other, each being the 



INTRODUCTION. 



tool for the other piece. Examples are the various 
kinds of spinning, weaving, and other textile machin- 
ery. All three forms are described in this volume, 
many examples being given among the pressure 
organs. 

47. It may appear from the preceding as if the 
theory of the action of the tool breaks through the 
logical arrangement given in the theoretical kinematics, 
since in Case a, one of the elements, the tool, cuts 
away and destroys its partner because it is enough 
harder to cut it. We must here distinguish between 
yielding and unyielding elements. This looks like a 
return to empiricism. The defect in the logic, how- 
is only apparent. All elementary pairs without excep- 
tion involve the idea that both of the partners evoke 
the latent forces by the action of deformation ; and at 
the same time the friction between the moving parts 
induces wear. Applied mechanics takes friction into 
account iii considering elementary pairs and investi- 
gates and provides for the consequent wear. The 
machine constructor endea-vors by all means within 
his power to reduce the alteration of form at points 
where it is not desired, but where it is the end to be 
accomplished he takes every opportunity to increase it. 
The forro.-changing action which occurs between the 
tool and the work-piece differs in degree only and not in 
kind, from the action taking place between the elements 
of every other pair in the machine [503]. 

48. A similar idea may arise in connection with 
the method of form-changing given above Under (b), 
in which an alteration of form takes place without an 
actual removal of any of the particles. In this case the 
the correspondence of the kinematic to the mechanical 
action is evident. In case 8, as already noted, the de- 
formation which takes place in non-rigid bodies makes 
it only practicable to obtain approximate solutions. 
This only involves a quantitative, and not a qualita- 
tive distinction [502]. 

Examples of this occur in the construction of in- 
struments of precision. It is not possible to construct 
even a simple cylindrical pair (case 19) such as a cen- 
tre for a theodolite, or for an astronomical telescope, 
entirely free from error. By the use of a variety of 
methods the errors are kept as small as possible, and 
then by other methods, nearly always kinematic, the 
residtial errors are determined and the proper correc- 
tions made. 

49. In other instances the designer may utilize the 
elastic yielding of the members of a kinemati: chain, 
as for instance in the method of Adolph Hirn, by 
which the springing of the beam of a steam engine is 
used to produce the indicator diagram of the steam 
pressure ; or the torsional deflection of a large shaft to 
measure the power transmitted.* 

This method is also found in Gidding's device for 
measuring valve friction (p. 285), and also in the 



* See Berliner Vtrkandlungen. 



Emery scale, in which a very small deflection of a 
diaphragm measures accurately weights of many tons. 

Although in many instances the deformations of 
material may be neglected, yet we should never per- 
mit ourselves to forget that they have been neglected. 
Otherwise important errors may creep into theoretical 
deductions, as well as in practical construction. This 
subject of the yielding of materials is receiving more 
attention at present than formerly. 

50. The "order" of a system of transmission is 
a subject of importance since there are several meth- 
ods by which the various parts may be kinematically 
arranged. I have applied the term "order" to the 
method of arrangement, and distinguish between three 
different methods. 

a. "Series Order." This " order " exists when a 
number of transmissions are arranged in series, so 
that each acts upon the following one. If in a single 
machine, two, three, four or "n" transmissions are 
thus arranged in series, I call the whole a system of 
the second, third, fourth or n*'> order. Examples are 
found in Figs. 766, 767. 

A transmission can return upon itself. This I have 
called a " ring " system of transmission. (See p. 208). 
This return to the original must always occur in the 
kinematic chain of any mechanism since the elements 
exist only in the relation of pairing (Case 5). In the 
system under consideration (Case 41), the groups of 
elements follow each other in a series, or line as it may 
be termed, whence I have termed such a series a 
" line " transmission (p. 257). Ring transmission may 
also be combined with line transmission, the line being 
divided into two or more parts. An example of the 
first kind is seen on page 229, in which the pump 
mechanism is combined with the steam mechanism, as 
a line with a ring system. An intermediate form be- 
tween ring and line transmission is referred to on page 
208. 

b. Combined Order. By this title is meant a com- 
bination of transmissions in which each transmission 
is connected to the next, but in which any one can be 
stopped without stopping the others. An example of 
this is shown in the ring transmission in Fig. 917. 

Under certain circumstances a number of the driven 
pulleys T^, T,, T^ - - Tn, may be allowed to run 
empty, in which case they become merely supporting 
sheaves (Case 43) ; as soon, however, as any load is 
thrown on any of them, the entire system is influenced 
by the increased stress upon the rope. 

Another example of "compo^md" order, is the 
multiple expansion steam engine. Here each engine 
of the compound, triple, or multiple expansion engine 
may be considered singly as a separate chain, and the 
entire machine as a series of transmissions. Each en- 
gine, Tj^, Tj, Tg, etc., exerts an influence upon the 
action of the others, but is not indispensable to their 
action, as would be the case if arranged in "series" 
order. Compound, Triple, Quadruple expansion en- 



XIV 



INTRODUCTION. 



gines are therefore, respectively of the second, third, 
and fourth order, but should also be considered to be- 
long to the class of "Compound order." 

c. "Parallel Order." This arrangement is the 
oldest and the one which occurs most frequently. It 
occurs when a number of different machines are all 
driven from one and the same transmission, this being 
the usual arrangement in manufacturing establish- 
ments. Any of the machines can be stopped or 
started independently of the others without affecting 
the motion, a suitable regulator being assumed. This 
principle may also be applied to the motors by which 
the transmission is driven, automatic couplings, such 
as shown on page loi, being used. A "parallel" 
order occurs in rope transmission when a number of 
ropes are used on the same pulley ; another instance 
is that of a train which is pulled by two locomotive 
engines. 

The three different "orders" are not always sharply 
defined, but the distinction' will be found of material 
assistance in the study of transmissions. An example 
in which all three, "orders" are used is found in the 
engine shown in diagram in Fig. 1023. Here the cyl- 
inder, piston, valve and steam form an escapement : 
the connection c 1 r being driven, and in turn operating 
a second r^ 1^^ b, and thence the valve. These three 
transmissions therefore form a "series" order, this 
also, returning to itself and being thus a ring system, 
and of the third order. The fly-wheel and its bearings 
form a dynamical power storage system, absorbing 
and giving out power in response to the irregularities 
of the action of the piston, this being of the ' ' com- 
pound " order. Frequently such an engine is made 
with an additional cut-off valve gear, with governor, 
also of " compound" order, also possibly a feed pump, 
("parallel" order) and the engine usually drives an 
extensive transmission system by which a number of 
machines are operated (" parallel " order). 

In § 260 is shown the manner in which physical 
and chemical trains are arranged in series, the action 
of heat, of gases and electricity being considered ; 
the steam engine being the most notable example. 

51. The magnitude of the exponent of the order 
of any train has an important influence upon the hurt- 
ful resistance of a machine, especially in a series order 
of a high degree. In such cases the injurious resist- 
ance increases at least directly as the exponent, and 
frequently more rapidly. It is therefore important in 
machine design to keep the degree of the order as low 
as practicable. In the system of pneumatic clocks of 
Mayrhofer (p. 171) the mechanism for several years 
was as high as the 17th order, but the degree subse- 
quently reduced to the 8th order. It may safely be 
affirmed that the simplicity of a machine may be 
measured by the closeness which the exponent of its 
order approaches unity. Examples are found in the 
Giffard injector, in which the guiding and driving 
mechanisms are united in one, and exponent becomes 



^i ; the same is true of Siemens Geyser pump, 
Fig. 971a. The apparatus of Morrison & Ingram, 
Fig. 1 181, is a device of the 2nd order, which acts by 
a combination of guiding and driving. 

52. The preceding pages have shown that applied 
kinematics, by means of the separation of the con- 
trolled motion into the forms of Guiding, Storing, 
Driving and Forming, and by means of the division of 
the various "orders, "has enabled the machine prob- 
lem to be solved as a whole. Theoretical kinematics 
has assisted in this solution by enabling the various 
problems to be investigated in a purely scientific man- 
ner. Without such a theoretical investigation, a sys- 
tem of applied kinematics would be an impossibility. 
At the same time practical instruction must be given 
by actual daily work as well. A clear understanding 
of the principles of the applied science cannot but be 
useful to the practical man, and as I believe, welcome 
also. 

The fundamental principles of machine construc- 
tion as I have sought to lay them down in the preced- 
ing pages, coincide in many points with the practical 
methods already in use. The practical mechanic is 
well acquainted with crank trains, gear trains, and 
the like, or if he is not familiar with them he is readily 
taught, but in combining these and arranging them so 
as to act upon each other the theory comes into play 
and shows clearly the best arrangement for the end in 
view. This is well shown in the case of the various 
valve gears, which have been in fact developed inde- 
pendently, instead of being the result of a theoretical 
analysis of various combinations of kinematic chains. 
The application of the kinematic analysis will facilitate 
work of this sort, making it clearer and simpler the 
more fully the fundamental principles are understood. 
For this reason I have introduced the kinematic princi- 
ples into this work, not to reduce invention to an art to 
be taught, but rather to bring the principles of science 
to its assistance. 

I am ready to admit that the general view of theo- 
retical kinematics which I have placed before the prac- 
tical man, may not be accepted without further proof 
being demanded. It may be considered only as an 
ingenious form of theorizing, of but little practical 
value. For the present I must ask my readers to prove 
by the test of practical application how far the princi- 
ples of kinematics may be made of genuine practical 
value. 

The principles included in cases 40 to 51 are 
practically applied in the latter half of this volume. 
The application of the analysis to the subject of ratchet 
gearing has produced an extensive series of results. 
Storage is clearly shown to be a form of ratchet gear; 
the discussion of the degree of "order" of ratchet 
trains will also, I believe, be found very useful. In 
the discussion of pressure organs (Chapter XXIII. and 
following) the subject of storage is highly developed. 
The notion of the two divisions of guiding, and driv- 



XIV 



INTRODUCTION. 



XV 



ing will also be found most useful. In like manner 
the methods of analysis as applied to ratchet trains, are 
found capable of equally prolific results when applied 
to pressure organ trains, not, to my knowledge, oth- 
erwise attainable. The great number of applications in 
this direction will be seen in § n},, these being the re- 
sult of the application of the theory sketched under 
Case 46, above. 

Since the subject of friction was considered in con- 
nection with rigid elements, it was also necessary to 
to take into account this resistance to the motion of 
fluids (§ 340), as also the loss of heat .in steam pipes 
(briefly discussed in § 338). In § 362 the very import- 
ant subject of boiler design is only generally consid- 
ered. 

The closing chapter relates to valves. These are 
treated as ratchets, not oniy from the theoretical 



standpoint, but also practically, and much more fully 
than in previous editions. The section on "fluid 
valves " will, I trust, be found of use to the practical 
man, as a subject worthy of further investigation. 

In closing, I may refer to the increasing size of this 
volume. In spite of my earnest efforts, it has not 
been possible to reduce its bulk. In many places 
evidence will be found of attempts at condensation, but 
nevertheless the work can hardly be called properly a 
"hand book" any longer. When discussing purely 
technical matters I can be brief, but in a practical work, 
it is above all things necessary to be clear and intelli- 
gible. In this I have endeavored to be guided by the 
dictum of Boileau : " Lhi ouvrage ne doit point para'lre 
trop travaille, niais il ne saurait Utre t?-op travailli.'' 
FuNCHAL, February, i88g. 

F. REULEAUX. 



T^BLE OF COIsrTE]SrTS. 



SECTION I. 

STRENGTH OK MATERIALS. 

Introductoi-y i 

Co-efficients of Resistance i 

Resistance to Tension and Compres- 
sion 2 

Bodies of Uniform Strength 2 

Resistance to Shearing 2 

Resistance to Bending 2 

Table of Sections 5 

Value of the Quantity S 8 

Sections of Uniform Resistance.... 8 
Bodies of Uniform Resistance to 

Bending 8 

Resistance to Shearing in the Neu- 
tral Plane 10 

Beams with a Common Load 11 

Resistance to Torsion 11 

Polar Moment of Inertia and Section 

Modulus II 

Bodies of Uniform Resistance to 

Torsion 13 

Resistance to Buckling 13 

Columns of Uniform Resistance. ... 13 

Compound Stresses 13 

Resistance of Walls of Vessels 15 

Calculation of Springs 18 

SECTION II. 

THE ELEMENTS OF GRAPHOSTATICS. 

Introductory 22 

Multiplication by Lines 22 

Division by Lines 23 

Multiplication and Division Com- 
bined 23 

Area of Triangles 23 

Area of Quadrilateral Figures 23 

Area of Polygons 24 

Graphical Calculation of Powers. ... 24 
Powers of Trigonometrical Func- 
tions 25 

Extraction of Roots 26 

Addition and Subtraction of Forces. 26 
Isolated forces in One Plane — Cord 

Polygon 26 

Equilibrium of External Forces of 

Cord Polygon 27 

Equilibrium of Internal Forces of 

Cord Polygon 28 

Resultant of Isolated Forces in One 

Plane 29 

Conditions of Equilibrium of Isolat- 
ed Forces 29 

Force Couples 29 

Equilibrium between Three Parallel 

Forces 30 

Resultant of Several Parallel Forces 31 

Decomposition of Forces 31 

Uniformly Distributed Parallel 

Forces 32 

Twisting and Bending Movements. - 33 

Determination of Centre of Gravity 33 

Resultant of Load on Water Wheel. 34 

Force Plans for Framed Structures. 35 

Force Plans for Roof Trusses 36 

Graphical Determination of Wind 

Stresses 37 

Force Plans for Framed Beams. ... 38 

Remarks 38 

SECTION in. 

THE CONSTRUCTION OF MACHINE 
ELEMENTS. 

Introductory 39 



CHAPTER I. 

RIVETING. 

Rivets 39 

Strength of Riveted Joints 40 

Table and Proportional Scale 40 

Riveting disposed in Groups 40 

Steam Boiler Riveting 42 

Table for Boiler Riveting 42 

Table of Weights of Sheet Metal ... 43 

Especial Forms of Riveted Joints... 43 

CHAPTER II. 

HOOPING. 

Hooping by Shrinkage 45 

Cold Hooping 45 

Examples of Forced Connections. . . 46 
Dimensions of Rings for Cold Forc- 
ing 47 

CHAPTER III. 

KEYING. 

Keyed Connections 47 

Cross Keyed Connections 48 

Longitudinal Keys 48 

Edge Keys 49 

Methods of Keying Screw Propel- 
lers 49 

Unloaded Keys 49 

Methods of Securing Keys 50 

CHAPTER IV. 

BOLTS AND SCREWS. 

Geometrical Construction of Screw 

Thread 50 

Whitworth Screw System 51 

Sellers' Screw Thread System 52 

Metrical Screw Systems 52 

New Systems 53 

Nuts, Washers and Bolt Heads 54 

Table for Metrical Bolts and Nuts . 55 

Weig;ht of Round Iron 55 

Special Forms of Bolts 55 

Wrenches 56 

Nut Locks 56 

Special Forms of Screw Threads. . . 58 

Screw Connections, Flange Joints. . 59 

Unloaded Bolt Connections 60 

CHAPTER V. 

JOURNALS. 

Various Kinds of Journals 60 

A. — LATERAL JOURNALS. 

Overhung Journals 61 

Example of Table of Journals 62 

Neck Journals 62 

Fork Journals 63 

Multiple Journals 63 

Half Journal 64 

Friction of Journals 64 

B. — THRUST BEARINGS. 

Proportions of Pivots 65 

Friction of Flat Pivot Bearings. ... 66 

Collar Thread Bearings 66 

Multiple Collar Thread Bearings. . . 66 

Compound Link as Thrust Bearing. 67 

Attachment of Journals 67 

CHAPTER VI. 

BEARINGS. 

Design and Proportion 68 

A. — LATERAL BEARINGS. 

Pillow Blocks 68 



Proportional Scale for Pillow Blocks 68 

Various Forms of Journal Boxes ... 69 

Narrow Bases— Large Pillow Blocks 69 

Adjustable Pillow Blocks 70 

Bearings with Three Part Boxes 70 

Wall Bearings 71 

Yoke Bearings 72 

Wall Brackets 72 

Hangers 73 

Adjustable Hangers 74 

Special Forms of Bearings 74 

B. — THRUST BEARINGS. 

Step Bearings 75 

Wall Step Bearings 75 

Independent Step Bearings 76 

Thrust Bearings with Wooden Sur- 
faces 76 

Multiple Collar Bearings 77 

Examples of Thrust Bearings 78 

CHAPTER VII. 

SUPPORTS FOR BEARINGS. 

General Considerations 79 

Simple Supports 79 

Multiple Supports for Bearings 80 

Calculation for Iron Columns 82 

Forms for Iron Columns 84 

CHAPTER VIII. 

AXLES. 

Various Kinds of Axles 85 

A. — AXLES WITH CIRCULAR SECTION. 

Simple Sy metrical Axles 85 

Non-Symmetrical Simple Axles 86 

Graphical Calculation of Simple 

Loaded Axles 86 

Proof Diagrams 87 

Axles Loaded at Two Points 78 

Railway Axles, Crane Pillars 88 

Axles with Three or More Bearings. 89 

Axles with Inclined Loads 90 

B. — AXLES WITH COMBINED SECTION. 

Annular Section 90 

Axles with Cruciform Section 90 

Modified Ribbed Axle gi 

Compound Axles for Water Wheels. 91 

Construction of Rib Profiles gi 

Wooden Axles , 92 

CHAPTER IX. 

SHAFTING. 

Calculations for Cylindrical Shafting 92 

Wrought Iron Shafting 93 

Line Shafting 93 

Determination of the Angle of Tor- 
sion 93 

Journals for Shafting — Round Rolled 

Shafting 94 

Combined Sections — Wooden Shaft- 
ing 94 

Shafting Subjected to Deflection, , . 94 

CHAPTER X. 

COUPLINGS. 

Various Kinds of Couplings. ....... 95 

I. Rigid Couplings 95 

II. Flexible Couplings 96 

Various Kinds of Flexible Couplings 96 
Couplings for Lengthwise and Par- 
allel Motion - 96 

Jointed Couplings 97 

III. Clutch Couplings 98 

Toothed Clutch Couplings., 98 



TABLE OF CONTENTS. 



xvn 



Friction Clutches 99 

Automatic Couplings loi 

CHAPTER XI. 

SIMPLE LEVERS. 

Journals for Levers loi 

Cast Iron Rock Arms 102 

Rock Arm Shafts 102 

Lever Arms for Rectangular Sec- 
tion 103 

Lever Arms for Combined Section . . 103 

CHAPTER XII. 

CRANKS. 

Various Kinds of Cranks 104 

Single Wrought Iron Cranks 104 

Graphostatic Calculation of Single 

Crank 104 

Cast Iron Cranks io5 

Return Cranks io5 

Graphostatic Calculation of Return 

Crank 105 

Simple Crank Axle 106 

Multiple Crank Shafts 107 

Locomotive Axles 107 

Hand Cranks 109 

CHAPTER XIII. 

COMBINED LEVERS. 

Various Kinds of Combined Levers, no 

Walking Beams no 

Scale Beams m 

CHAPTER XIV. 

CONNECTING RODS. 

Various Parts of Connecting Rods. 112 
Connections for Overhung Crank 

Pins 112 

Stub Ends for Fork Journals 114 

Connections for Neck Journals 114 

Round Connecting Rods 116 

Rods for Rectangular Section 116 

Channeled and Ribbed Connecting 

Rods 117 

Cast and Wrought Iron Rods nS 

CHAPTER XV. 

CROSS HEADS. 

Various Kinds of Cross Heads 118 

Free Cross Heads 119 

Cross Heads for Link Connections.. 119 

Cross Heads for Guides 119 

Guides and Guide Bars ■ 121 

CHAPTER XVI. 

FRICTION WHEELS. 

Classification of Wheels. 122 

The Two Applications of Friction 

Wheels 123 

Friction Wheels for Parallel Axes . . 123 

Friction Wheels for Inclined Axes- ■ 124 

Wedge Friction Wheels 125 

Special Applications of Friction 

Wheels 126 

Roller Bearings 126 

CHAPTER XVII. 

TOOTHED GEARING. 

Classification of Gear Wheels 127 

A. The Construction of Spur 

Teeth 

General Considerations 12S 

Pitch Radius, Circumferential Divis- 
ion 128 

Table of Radii of Pitch Circles 128 

General Solution of Tooth Outlines 129 

The Action of Gear Teeth 129 

The Cycloidal Curves 130 

The Generation of Cycloidal Curves 130 
Tooth Outlines of Circular Arcs. . . . 131 
Evolute Teeth for Interchangeable 

Gears 131 

Pin Teeth 132 

Disc Wheels with Pin Teeth 133 

Mixed Tooth Outlines, Thumb Teeth 133 

Tooth Friction in Spur Gearing 134 

General Remarks .'i 135 

B. Conical Gear Wheels 



General Considerations 135 

Construction Circles for Bevel Gears 135 
The Plane Gear Wheel 136 

C. Hyperboloidal Gear Wheels 
Base Figures for Hj'perboloidal 
Teeth for Hyperboloidal Gears 138 

D. Spiral Gears 

Cylindrical Spiral Gears 13S 

Approximately Cylindrical Spiral 

Gears 139 

Spiral Gear Teeth and their Friction 140 

Spiral Bevel Gears 141 

Globoid Spiral Gears 142 

E. Calculation of Pitch and 

Force of Gearing 

Pitch of Gear Wheels, Tooth Sec- 
tion 144 

Pitch and Face of Hoisting Gears. . 144 
Table of Cast Iron Hoisting Gears. 145 
Pitch and Face of Gearing for Trans 

mission 145 

Examples and Comments 147 

F. Dimensions of Gear Wheels 

The Rim 147 

The Arms of Gear Wheels 149 

Table of Gear Wheel Arms 149 

Gear Wheel Hubs 150 

Weight of Gear Wheels 150 

CHAPTER XVIII. 

RATCHET GEARING. 

Classification of Ratchet Gearing. . . 150 
Toothed Running Ratchet Gears. . • 150 

The Thrust upon the Pawl 152 

The Sliding Flanks 153 

Spring Ratchets, Quadrants 153 

Methods of Securing Pawls, Silent 

Ratchets 153 

Special Forms of Ratchet Wheels. . 154 

Multiple Ratchets 154 

- Step Ratchets 155 

Stationary Ratchets 156 

Ratchets of Precision 157 

General Form of Toothed Ratchets. 158 
Dimensions of Parts of Ratchet 

Gearing 158 

Running Friction Ratchets 15S 

Release of Friction Pawls i5i 

Stationary Friction Ratchets 161 

Releasing Ratchets 162 

Checking Ratchets 163 

Continuous Running Ratchets 164 

Continuous Ratchets with Locking 

Teeth 165 

Locking Ratchets 166 

Escapements, Their Varieties 167 

Uniform Escapements 167 

Periodical Escapements 169 

Adjustable Escapements 170 

General Remarks upon Ratchet 

Mechanism 171 

CHAPTER XIX. 

TENSION ORGANS CONSIDERED AS MACHINE 
ELEMENTS. 

Various Kinds of Tension Organs.. 172 

Methods of Application 172 

Technological Applications of Ten- 
sion Organs 177 

Cord Friction 177 

Ropes of Organic Fibres 178 

Wire Rope 179 

Weight of Wire Rope and its Influ- 
ence 180 

Stiffness of Ropes 181 

Rope Connections and Buffers 181 

Stationary Chains 182 

Running Chains , 182 

Calculations for Chains 183 

Weight of Chain 183 

Chain Couplings 1S4 

Chain Drums and Sheaves 185 

Ratchet Tension Organs 1S5 

CHAPTER XX. 

BELTING. 

Self-Guiding Belting 186 

Guide Pulleys for Belting 1S6 



Fast and Loose Pulleys 188 

Cone Pulleys iSg 

Cross Section and Capacity of Belts. 190 

Examples of Belt Transmission 191 

Belt Connections 191 

Proportions of Pulleys 193 

Efficiency of Belting 194 

CHAPTER XXI. 

ROPE TRANSMISSION. 

Various Kinds of Rope Transmis- 
sion ig4 

A. Hemp Rope Transmission. 194 
Specific Capacity, Cross Section of 

Rope 195 

Sources of Loss in Hemp Rope 

Transmission 195 

Pressure and Wear on Hemp Rope. 196 

B. Cotton Rope Transmission. 196 

C. Wire Rope Transmission.... 196 
Specific Capacity, Cross Section of 

Rope ig6 

Influence of Pulley Diameter 197 

Deflection of Wire Ropes 198 

Tightening Driving Ropes 200 

Short Span Cable Transmission.... 200 
Transmission with Inclined Cable.. 200 
Construction of the Rope Curve. . . . 202 

Arrangement of Pulleys 202 

Construction of Rope Pulleys 202 

Construction of Pulley Stations.... 204 
Efiiciency of Rope Transmission . . . 205 
Reuleaux's System of Rope Trans- 
mission 206 

CHAPTER XXII. 

CHAIN TRANSMISSION. STRAP BRAKES. 

Specific Capacity of Driving Chains 211 
Efficiency of Chain Transmission. . . 213 
Intermediate Stations for Transmis- 
sion 213 

Strap Brakes 214 

Internal Strap Brakes 316 

CHAPTER XXIII. 

PRESSURE ORGANS CONSIDERED AS MA- 
CHINE ELEMENTS. 

Various Kinds of Pressure Organs. 216 
Methods of Using Pressure Organs. 216 

Guiding by Pressure Organs 216 

Guide Mechanism for Pressure Or- 
gans 217 

Reservoirs for Pressure Organs.... 2i<8 
Motors for Pressure Organs 2 it) 

A. Running Mechanism for 

Pressure Organs 

Running Mechanism operated by 

Weight 219 

Running Mechanism Operated by 

Impact 219 

Running Mechanism Operated 

against Gravity 221 

Running Mechanism in which the 

Motor is Propelled 222 

B. Ratchet Mechanism for Pres- 

sure Organs 

Fluid Running Ratchet Trains 223 

Fluid Trains with Stationary Rat- 
chets. 225 

Escapements for Pressure Organs.. 226 

A. Unperiodic Escapements for 

Pressure Organs 

Fluid Escapements for Transporta- 
tion 227 

Hydraulic Tools 228 

Pressure Escapements for Moving 

Liquids 228 

B. Periodical Pressure Escape- 

ments .... 

Pumping Machinery 229 

Fluid Transmission at Long Dis- 
tance 233 

Rotative Pressure Engines 233 

Valve Gears for Rotative Engines . . 234 

C. Adjustable Power Escape- 

ments 

Adjustable Pump Gears 236 



xvm 

Adjustable Gears for Rotative Mo- 
tors 237 

D. Escapements for Measure- 
ment of Volume 

Running Measuring Devices 239 

Escapements for Measurement of 

Fluids 239 

Technological Applications of Pres- 
sure Organs 240 

CHAPTER XXIV. 

CONDUCTORS FOR PRESSURE ORGANS. 

Formulae for Cast Iron Pipes 242 

Weights of Cast Iron Pipes 242 

Pipes for High Pressures 242 

Wrought Iron and Steel Pipes 243 

Steam Pipes 245 

Pipes of Copper and other Metals . . 246 

Resistance to Flow in Pipes 246 

Methods of Connecting Cast Iron 

Pipes ^48 

Connections for Wrought Iron and 

Steel Pipes 249 

Connections for Pipes of Lead and 

other Metals 251 

Flexible Pipes 252 

Pistons 252 

Plungers and Stuffing Boxes 253 



TABLE OF CONTENTS. 

pistons with Valves 255 

Piston Rods 255 

Specific Capacity of Pressure Trans- 
mission systems 255 

Ring System of Distribution, with 

Pipe Conductors 256 

Specific Capacity of Shafting 257 

Specific Value of Long Distance 

Transmissions 259 

CHAPTER XXV. 

RESERVOIRS FOR PRESSURE ORGANS. 

Various Kinds of Reservoirs 260 

Cast Iron Tanks 260 

Riveted Tanks 260 

Tanks with Concave Bottoms 262 

Combination Forms for Tanks 264 

High Pressure Reservoirs, or Ac- 
cumulators 264 

Steam Boilers 265 

Boiler Details subjected to Internal 

Pressure 266 

Boiler Flues subjected to External 

Pressure 269 

Future Possibilities in Steam Boiler 

Construction 270 

Reservoirs for Air and G-as 272 

Other forms of Storage Reservoirs. 273 



CHAPTER XXVI. 

RATCHETS FOR PRESSURE ORGANS, OR 
VALVES. 

The two Divisions of Valves 273 

A. Lift Valves 274 

Hinged or Flap Valves 274 

Round Self- Acting Valves 275 

Unbalanced Pressure on Lift Valves 277 
Closing Pressure of Self-Acting 

Valves ■ • ■ , . 278 

Mechanically Actuated Pump Valves 27S 

Valves with Spiral Movement 279 

Balanced Valves 279 

B. Sliding Valves 

Rotary Valves and Cocks 281 

Gate Valves for Open and Closed 

Conductors 282 

Slide Valves 282 

Balanced Slide Valves 285 

Fluid Valves 287 

Stationary Valves 289 

Stationary Machine Elements in 

General 2S9 

SECTION IV. 
Mathematical Tables 291-301 



ERI^^T^. 



Page 14, Case IV., first panel of table should read P=4 it'^ ^ 

Page 15, line 13 from bottom, second column, omit words "%" thick." 

Page 53, line 31 from bottom, second column, read "interpolated diameter/' instead of "interpolated meter." 

Page 61, formula (89) substitute P, instead of p. 

Page 61, line 11 from top, second column, after "Proportions of Journals," insert the formula number (93). 

Page 63,, line 39 from top, first column, after "Formula for Fork Journals" insert the formula number (98). 

Page 64, the formulae on lines 12 and 14, of § 96, should be numbered respectively (99) and (100). 

Page 64, line 33 from top, second column, for " Prow^ny " read "Prony." 

Page 89, line 17 from bottom, first column, for 85 mm.^8%" read 85 mm. =3%". 

Page 89, illustration at the bottom of second column, the diagram to the left should be Fig. 409, and that to 
the right. Fig. 410. 

Page 97, line 16 from bottom, second column, for "drawn '' read "driven." 

Page 103, the last formula on first column should be numbered (154) instead of (155). 

Page 144, formula at bottom of first column, the cube root sign applies to the vifhole of the second member 
and not to the numerator only, as printed. 

Page 175, line 17 from bottom, second column, for Harturcn, read Hartvi'ich. 

Page 195, line 29 from top, second column, for " can only be given by indeterminate results," read " can 
only give approximate results." 

Page 206, title of § 301 read Reuleaux's instead of Reuleux's. 

Page 255, example in second column, for 4 in. stroke, read 40 in. stroke. 

Page 263. The following revisions of formulas (385) and (386) have been commumcated bv Prof. Reuleaux 
and should be inserted : 



2 ( 

R 
■Tg^Y— 






■C385) 



•(386^ 



THE CONSTRUCTOR: 

A HAND-BOOK OK MACHINE DESIGN. 

BY K. REULEAUX. 



Section L— STRENGTH OF MATERL\IvS. 



Introductory. 

The study of the strength of materials ultimately depends upon 
the question of the resistance which rigid bodies oppose to the 
operation of forces, and the following definitions are to be 
noted : 

SuPERFiciAi^ Pressure is the pressure upon a unit of surface. 

Tensii,E Strength is the resistance per unit of surface, 
■which the molecular fibres oppose to separation. 

MoDUi,us OF RESIST.4.NCE is the strain which corresponds to 
the limit of elasticity, compression and extension, each hav- 
ing a corresponding modulus. 

Modulus of Rupture is the strain at which the molecular 
fibres cease to hold together. 

Modulus of Elasticity is the measure of the elastic exten- 
sion of a material, and is the force by which a prismatic body 
would be extended to its own length, supposing such extension 
were possible. 

Theoretical Resistance is the force which, when applied to 
any body, either as tension, compression, torsion or flexure, 
will produce in those fibres which are strained to the greatest 
extent a tension equal to the modulus of resistance ; or, in 
other words, it is the load which strains a body to its limit of 
elasticity. 

The Practical Resistance often improperly termed merely 
Resistance, is a definite but arbitrary working strain to which a 
body may be subjected within the limits of elasticity. 

The Coefficient of Safety is the ratio between the theo- 
retical resistance and the actual load, or, what amounts to the 
same thing, the ratio between the elastic limit and the actual 
tension of the fibres. 

The Breaking Lo.\d is that load which causes in those fibres 
■which are subjected to the greatest strain, a tension equal to 
the modulus of rupture ; in every case this is equal to the force 
necessary to tear, crush, shear, twist, break, or otherwise de- 
form a bod}'. 

The Factor of Safety is the ratio between the breaking load 
and the actual load. 

As a general rule, for machine construction, the Coefiicient 
of Safety may be taken as double that which is used for con- 
struction subjected to statical forces. Circumstances may also 
require it to be taken as either greater or less than the custom- 
ary value, sometimes even narrower than is permitted for stati- 
cal forces. Care must be taken never to permit a material to be 
strained in use to its theoretical resistance ; although, indeed, 
there are some materials, such as wrought iron, which have 
been strained slightly beyond the limit of elasticity, without re- 
ducing the breaking load, or causing any apparent injury. (See 

The determination of the breaking load, and consequently 
the use of the modulus of rupture, is limited to those cases in 
which the actual breaking of the structure must be considered ; 
but for the actual calcvdations of working machinery the modu- 
lus of resistance, or limit of elasticity is of primary importance. 

?2. 

Coefficients of Resistance.* 
The coefficients given in the following table are selected as the 
mean of man}' experiments upon the various materials named. 
Under the title " Wood ' ' is given an average value from ex- 
periments made with oak, beech, fir and ash. Those materials 
which show the greatest difi'erence between the modulus of 
rupture and the limit of elasticity also possess in the highest 
degree the property of toughness. 

* Throughout the original work all dimensions and quantities aie given 
in the inetric system, but these have been transformed into English units 
for English readers, except in the following table, where both are given. 
— Trans. 



Experiments upon wrought iron show that a strain beyond 
the limit of elasticity, if not carried too far, although it will 
cause a permanent deformation, will not lower the modulus of 
elasticity, but will raise the modulus of resistance. 

For example, a rod of wrought iron, subjected to a tensile 
strain of 28,400 lbs. per square inch, was subsequently found to 
have its limit of elasticity raised from 21,300 lbs. to 28,400 lbs. 
(This property is utilized in drawing wire). 

Tenacity is a particularly desirable property for a material 
of construction, and it may generally be approximately meas- 
ured by the ratios K : T and Ki : Tj. 

If the rod above referred to be subject to compression it 
will return to its former limit of elasticity. 

Table of Coefficients.* 



M-iTEEIAL. 



Wrought Iron 

Iron Wire 

Sheet Iron 

Cast Iron 

Spring Steel (hardened) 
Cast Steel (not hardened) 
Cast Steel (spring temper) 
Copper (hammered) . 

Copper Wire 

Brass 

Brass Wire 

Bell Metal (bronze) . . 
Phosphor Bronze . . . 

Aich Metal 

Lead 

Wood 

Hemp Rope (new) . 

Hemp Rope (old) . . 

Belting 

Granite 

Limestone 

Quartz 

Sandstone 

Brick 

Limestone Masonry 

Sandstone Masonry 

Brickwork 





Modulus of Re- 


Modu 


us of 




sistance. 


Rupt 


ure. 


Modulus 
of 


















plasticity. 




Com- 


Ten- 
sion. 


Com- 


E. 


Tension. 
T. 


pres- 


pres- 






K. 


sion. 






Tj. 


Kj. 


20,000 


15 


IS 


/; 


22 


28.400,000 


21,300 


21,300. 


56,800 


3",24a 


20,000 


30 




70 




28,400,000 


42,600 




99,400 




17,000 






32 




24,140,000 






45,440 




10,000 


7.5 


15 


11 


63 


14,200,000 


10,650 


21,300 


15,620 


88,460 


20,000 


50 to 70 




80 




28,400,000 


70 to 90,000 




113,600 




20,000 


25 




So 




28,400,000 


35,500 




113,600 




30,000 


65 to 150 




100 




42,600,000 


go to 200,000 




142,000 




11,000 
15,620,000 


2.5 

3,550 




30 

42,600 


70 
99,400> 


13,000 


12 




40 




18,460,000 


17,040 




46,800 




6,500 


4.S 




12 


no 


9,230,000 


6,Si6 




17,040 


i56,2oo> 


10,000 


13 




50 




14,200,000 


18,460 




71,000 




3,200 


9 




13 




4,544,000 


12.7S0 




18,460 






15 

21,300 




36 

51 120 






15 
21,300 





75 
106,500 




500 


I 




11 


5 


710,000 


1,420 




1,846 


7,100 


11,000 


2 


1.8 


9 


5 


1,562,000 


2,840 


2,556 


12,780 


7,100 


250 (?) 


5(?) 




12 




355,000 


7,100 




17,040 




5o{?) 


M?) 




5 




71,000 


1,420 




7,100 




15 to 20 


1.6 




2.9 




20 to 30,000 


2,272 




4,iiS 


8 
11,360 

5 
7,100 






























17,040 

7 
9,940 



















0.6 










85» 










7,100 
1-5 










=,130 
0.4 










56S 



* The upper figures are kilogrammes per square millimetre, and the 
lower figures are pounds per square inch. 



THE CONSTRUCTOR. 



?3. 
Resistance to Tension and Compression. 

A body is said to be under tension when the action of a force 
P, tends to extend it in the direction of its length. When the 
force acts in the opposite direction the body is said to be under 
compression ; but when the length is great in proportion to the 
cross section, a combined action occurs. (See ? i6.) 

I^et (7 be the cross section of the member : S, the strain due to 
the action of the force P ; then if we neglect the weight of the 
material we have : 



P=Sq 



(I) 



*EJ^aniple. A rafter exerts a horizontal thrust of 22,000 pounds, which 
must be borne by a rod of circular cross section, li we make 5" = 7100 
pounds we have for the diameter of the rod d. 



Sq~- 



from which d ^= i.c 



7100 — d- = 2 
4 



The principal action which the application of a force to a 
member produces is the consequent elongation or compression. 
A prismatical body subjected to the action of a force P, will 
have its original length / increased by in amount A, determined 
by the formula 






(2) 



and this holds good as long as 5 is not greater than the modu- 
lus of Resistance for tension T. This relation is also true for 
compression, in which case the limit depends upon the modu- 
lus of resistance T^ for compression. 

Example. Suppose the rod. whose diameter was determined in the pre- 
ceding example, to have a length of 114 ft. 10 in. or 137S inches, its elonga- 
tion under those conditions would be 

,^_ i378 X 7'°° ^ 
28,400,000 

The preceding formula (2) is a fundamental one, and upon it 
is constructed the whole systematical study of the strength of 
materials. 

Formula (i) is of use when a section is strained beyond the 
limit of elasticity, as by it we may determine the force' required 
to rend or crush a material, using the proper Modulus of Rup- 
ture. 

Example. The force necessary to pull the above given rod 
asunder is 

P=Kq 

IT 

or /'=56,Soo X (2)' — = 178,442 lbs. 



I A. 
Bodies of Uniform Strength. 

By bodies of uniform strength are meant those in which the 
shape is so made that the cross sections at various points are 
subjected to the same strain S, and consequently a proportion- 
ally economical distribution of material secured. 

Such forms are not often employed in practice, although ap- 
proximate shapes may often be adopted, but they serve in many 
cases to determine the general style of a structure, and give it 
the effect of proportional strength without adhering too closely 
to the exact form. These forms will be found of value to the 
designer for both reasons : principally as a guide to the style of 
his work rather than for close determinations of economy. 

If a designer has become thoroughly familiar with the resist- 
ing capacity of various shapes, and can keep them so clearly in 
his mind that he can perceive the general form of the proper 
curve to be used in any particular case, he will be able to pro- 
duce, with an artistic freedom, designs which will approach the 
shapes indicated by mathematical analysis. 

The following forms are alike suitable for tension and com- 
pression. As examples of their practical use, the first two are 
applicable to cast columns, and .the third is suitable for chim- 
neys of masonry as well as for high piers of bridges and via- 
ducts. 



*In all cases the quantities given in the original examples have been 
converted into their English equivalents, which will account for the un- 
usual quantities chosen. {Trans.). 



SHAPE. 





EQUATION. 



REMARKS 



4=V 



;; 



p 
5 



cf 




p 



— X 

.s 



e = 2.718 = Base of 
natural logarithms. 

P 
log ^ = log — -1- 



o-434^-^ 



P, is distributed un- 
iformly throughout 
the whole length of 
the figure. Cross sec- 
tion circular. Profile 
parabolic. Approxi- 
mate form, a trun- 
cated cone with end 

diameter = — 
2 



P, is uniformly de- 
creased from above 
downward. Cross 
section circular. 
Form conical. 



The body is strain- 
ed by its own weight, 
y being the weight 
of a unit of volume. 
The cross section in- 
creases with the in- 
creasing load in the 
logarithmic propor- 
tion given. 



is- 
resistance to shearing. 
A body is said to be subjected to a shearing strain in any- 
cross section when the distorting force acts in the plane of that 
cross section. 

Let q, be the sectional area, and S, the force acting upon it, 
so that we have as in the case of tensile and compressive 
strains 

^=S^ (3) 

The limit of elasticity will be reached when 5=5 0/ the lesser 
of the two Moduli of Resistance of the material, in the case of 
wrought iron, where T = Ti = 21,300, S = 17,040 lbs. while for 
cast iron T<T, and ^10,650 and S = 8,520. In this case the 
maximum strain is not in the plane of the cross section but at an 
angle of 45° with it, and is f the value of S. The deformation 
which two surfaces suffer under a shearing strain is very slight 
within the limits of elasticity, but it is noticeable in the case of 
torsional distortion where numerous sections under shearing 
strain adjoin each other. 

Equation (3) holds good for cases in which the division of 
the adjoining surfaces occurs, such as shearing, planing, bor- 
ing, in fact, the work of the entire class of machines which act 
by the removal of a portion of the material upon which they 
are working. 

The strain of separation S, in this case is somewhat less 
than the modulus of rupture for tension (K). This is due to 
the fact that in the case of shearing forces both K, and K, 
come into action. For the calculation of such machines a 
coefiBcient of rupture equal to i.i K may be taken. 

16. 

Resistance to Bending. 

Elasticity and Strength of Flexure. 

A prismatic body upon which the external forces act in a 

direction at right angles to the axis of the prism, is said to be 



THE CONSTRUCTOR. 



subject to strains of flexure. As long as the elastic limit is not 
overstepped there will exist an equilibrium in each normal 
section of the prism between the moments of the external 
forces on the one hand and the moments of the opposing 
internal forces on the other hand, both acting about the neu- 
tral axis of the given section. This may be considered as a 
sort of equator of each section since it passes through its cen- 
tre of gravity at right angles to the plane in which the bending 
takes place. It thus divides the section into tw-o portions, in 
one of which all the fibres which are parallel to the axis of the 
prism are subiected to a tension proportional to their distance 
from the neutral axis (the tension side of the section), while in 
the other portion the corresponding fibres are subjected to 
compression in 'a like proportion (the compression side of the 
section). It follows that fibres which are at equal distances 
from the neutral axis will be deformed to the same extent. 
The resistance to bending is therefore a combination of the 
resistances to tension and to compression, both acting in a 
peculiar manner, that is, in rotation about an axis. 
Let : 

i)/:=the statical moment of a cross section subjected to a 

bending force, taken with reference to the neutral axis 

of the section, that is, the so-called moment offeree. 
y=^the moment of inertia of the section with regard to 

the neutral axis, 
a^the distance from the neutral axis of the fibres which 

are subjected to greatest tension or compression ; 

7. e. , those which are farthest from the neutral axis. 
5':=the corresponding strain in these fibres, then : 



^J: 



a 



(4) 



The product 6" — is the statical moment of the entire collec- 
tion of fibres of the cross section with reference to the neutral 
axis and is the strain moment of the section under considera- 
tion. For any prismatic shape subjected to a bending force P, 
whose lever arm is x, we may put M^=Px- for each cross sec- 
tion, and for each section this will have a different value. The 
section at which the product Px has its greatest value is the 
section of danger, and the bending force P, which produces in 
it the strain S, is the resisting strength of the shape for the 
strain S, so that in this case we have 

SJ 



in which ,r,„, is that value of x, which makes Px a maxi- 
mum. 

The axis passing through the centres of gravity of succes- 
sive cross sections of a figure is not subject to any change 
of length under flexure, but is only curved, and to find its 
radius of curvature under an}' load we have the formula 



■ M 



(6) 



This curve is called the elastic curve, and is determined by 
the general equation 






(7) 



P=- 



XmCl 



(5) 



In the following tables are given the values of the quan- 
tities for calculations of fiexure under the various conditions 
shown in the figures, being : 

The Bending Moment 3T, for any point x, 

The Bending Load P, according to formula (5), 

The Co-ordinates x, y, for the elastic curve. 

The value, _/, of the abscissa r, at the point of application of 
the force as shown in Figures I to VI, and the value of the 
greatest deflection /, in the cases shown in Figures VII to 
XIII. 

In all the examples the weight of the beam itself has been 
neglected, as this may usually be done in machine construc- 
tion, although not in bridge work. 

Figures VII to X are suitable for the latter purpose, as 
in them the weight of the beam may readily be taken into 
account. 

In Figures XI and XII may be seen how unequal distri- 
bution of the load affects the sustaining power, as a beam 
loaded like Fig. XI or XII has i j-2 times the sustaining power 
of one loaded like VII or VIII, or for the same load a corres- 
pondingly reduced deflection. These are important considera- 
tions in connection with the distribution of weights in buildings. 
The distribution in XIII is also unfavorable, as it has only 3^ 
the sustaining power of VIII, with a greater deflection. 

It is to be noticed that the deflection/, increases as the third 
power of the length, and that it varies greatly under the various 
conditions given. 



Example. 





Bending 
Moment M. 



M = Px 



M = 



Px 




Vor A C 

M = -5- P:t 
16 



For .5 C 



U2 16 / J 



Sustaining 
Power P. 



SJ 

la 



P= 4 



SJ 



■ / SJ 
c C\ a 



p_ 16 SJ 

3 la 



Equation of the Elastic Curve. 



^ JE 



'LT— i- —1 



, p 

JE 



/3 r j: i ^1 

16 L ' ~ 3 '^J 



P (fci- 

' J E e 



lie ci c"- CiJ 



P c-ci 
JE 






P 
JE 3 



32 L / 3 pJ 



JE 32 L+ I 

II Xj'-] 
3 I' J 



Deflection 7^ 



._ P P 
<^ JE 3 



/ = 



JE 



48 



JE 3 i- 
f=^ maximum when 






^/^TT 



^=^ 



7^ 



ymax 



-fh- 



Pl^ 



5 48 /£ 
■when jr = / y ^ 



Remarks. 



Weakest Sec- 
tion at B. 



Weakest Sec- 
tion in the 
Middle. 



Weakest Sec- 
tion at C 

Reactions 



^^7 



Weakest Sec- ' 
tion at B, 



Reaction at ^ 



THE CONSTRUCTOR. 



Example, 





* ""7 i 


j 


V: r ^ ^ 


-■. 


(§ 



Bending Moment M. 



M = 



P/ /^ J. \ 
2 W ~ 4-/ 




Sustaining 
Power i'. 



P = 



11 

la 



Equation of Elastic Curve. 



J£ 



l6 L ^ 3 '■•' J 



ca 



G-1 + ^] 



^^^=M;-T + 



I .r . 2 A-- 



-a-i?) 



/a 



la 



P= 8 Il- 
ia 



SJ 



P= i 



SJ 



SJ 



r = 6 



SJ 

la 



= J- P + )/ p"--^' + I (jr^L\ 



iu which 

P = l^ 
Fc 



Deflection y! 



f~JLJL 
JE 193 



P P a 
JS S I 



P 
JE 



ill ^ ii\ 



JE 2 



4 L/ p ^ i*\ 



y = 



JE 






■y£ 24 






^ y£ 12 Yi 5 ^5 J 



^ ~ JE 12 Ls / ./3 ■*" /4 s '^ J 



^ ~ JE 12 Ls ^ -'='5 "^0 



/■ 



JE 8 



p 5/3 

JE 384 



/= _fL JL 

JE 192 



-JLJL. 

'jE 384 



V£ 15 



V^ 320 



/= 



JE 60 



Remarks. 



Weakest Sections at .fi 
and C. 



Weakest Section at an in- 
determinate point be- 
tween A and B. 



Weakest Section at B. 



Weakest Section in the 
Middle. 



Weakest Section at C 
Greatest Deflection when 



x/al) 



Reaction at X= % P 
Turning point x =^l 



Weakest Section at B. 
Point of reversal at 



Weakest Section at B. 



Weakest Section in the 
Middle. 



Weakest Section in the 
Middle. 




XIV. In the case of a beam supported upon two symmetrically placed supports A and B, and carrying a 

P X f X c\ 

uniformly distributed load P, we have for the bending moment ISI = ['-. I H j. 

The supporting power varies according to the position of the supports, and also with the relation of c to /; 
it will become a maximum when c = 0.207 ^ 1 that is, / ( */ J I 



THE CONSTRUCTOR. 



The supporting power will tiien approximate to 

la ' 



P=A7'^ 



or nearly six times as great as in case VIII, showing 'the ad- 
vantage of this method of support. The weakest sections are 
at A, B and C. 



J 



§7. 

Table of Sections. 



The value of^ in equation (4) depends almost entirely upon 

the shape of the cross section of the beam, and this we shall 
hereafter call the Section Modulus. 

The following table shows a large number of sections in use 
for various purposes, and gives the corresponding values of the 
following quantities : 

The equatorial moment of inertia J, for the neutral axis, 
shown in the figures by the dotted line. 

The greatest distance a, of the fibres under tension and com- 
pression, or their separate values a' , and a"/ when the section 
is not symmetrical about both axes. 

The equatorial section modulus Z = — , for which two values 

are given, when a', and a'\ are different ; and 

The sectional area of the figure, which will be found of 
service in calculating weights. 

To determine the value of a, experimentally or graphically, 
a model of the section may be cut out of cardboard, and its 
centre of gravity found by balancing on knife edges, or else the 
graphostatic method given in ? 46 may be employed. 

The following example will serve to show the application of 
the table : 

Example. Required the moment of inertia of a circular section 4 inches 
in diameter. According to No. XX in tlie table ; 



^-T,'''- 



■■ 0.0491 d^ = 0.0491 X 256= 12.5706 



By making various combinations of the forms given in the 
tables other sections may be obtained to which the same 
formula will apply. As an example, the Section No. VIII may 
also be used for a rectangular tube, and No. XI for an E shaped 
section. 

It is a matter of some importance for the designer to keep in 
mind some general conclusions, which may be drawn from the 
tables as to the influence of various shapes upon the strength. 
It will be plainly seen that the depth of a section is the dimen- 
sion which has the greatest influence upou the strength, and 
also that those portions of the section which are furthest re- 
moved from the neutral axis are of the most service. 

It is upon this point that the peculiar strengthening effect of 
ribs depends, and which makes their use so advantageous in 
cast iron constructions. These ribs do not act so much by the 
mere strength of their own cross section as by the fact that 
they strengthen those portions which are furthest from the 
neutral axis. This is a feature to be carefully watched, and its 
importance may be made clear by an example. 

If we take a section of the form given in No. XV., and 
make its dimensions as follows ; b ^ ^ bi ,/! = 12 bi , /n ,=: 11 
bj (Fig. I, § 9) and then divide it into two rectangular parts by 
a horizontal section, we have for the modulus of each section : 



II' X V_ 



20j4 1^1' and 



8 V 



which, together, give 21.5 5,'. 

The same material, when taken as a whole, in a single sec- 
tion (see § 9) would have a modulus Z = 34.8 bi^, so that it 
has more than i]4 times the resistance of its separated por- 
tions, and as a matter of fact the right angle rib or T head is 
about ten times the value in that connection than if taken by 
itself. This is also found in a still higher degree in sections of 
other shapes. 



SECTION TABLE. 



Section. 



Moment of Inertia^. 



Distance ( 



Section Modulus 2. 



*-b-*i 




iA» 






bh 





g 


^ 




^4 


Ih 
— t- — - 



i(kt—h-p) 



III. 



"Ei 



....*....» 



r^* 



IV. 




ft:^^ 




i6 



5413 '5* 



,VI. 




5\/J 
16 



b 



b Jl- = oM6b 
4 



i (A3 — h^S) 
6h 



H,h — hi) 



6 



y^t 



=0.118 3» 



l-^» 



3\/3 



3» = s 



16 



av^s 



THE CONSTRUCTOR. 



SECTION TABLE— (CiB^/TOW^f)- 



Section. 



Moment of Inertia y. 



Distance a. 



Section Modulus Z. Area F. 



VII. 






VIII. 






IX. 



X. 



rc-'b--»i 



XI. 




XII. 



"T'iiiH!- 




XIII. 




XV. 




i^" 



XVI. 



XVII. 



> *- 



C 4- 2 \/2 



I* = 0.638 .^ 



i/iS—{i — d{)Ai3 



^ (/i3 — /tl3) + ^^l (/;i3 — /jqS) 



*/i3 + ii/ji! 



ili> — (i — i^)!l-fi + iih^^ 



i m + (/ii — b) h^ +(h — hi) 



36 



36(^ + ^1) 



/l3 



l[^3(a'3-/3)+^,(/3 + a"3)] 



0.9243 



0.677 ^ 



2.828^2 



3 Ifi — (6 — ^0 /;i3 
6/; 



bh—(h — bi)h-i 



b {Jfi — h-f) + ^1 {h-f — /z„3) 
&I1 



b(h-h{) + bi{h-,-h^ 



b B + ^1 h{' 
oh 



ih+iihi 



b tfi —ib — i.,) h-f + bj hj' 
th 



bh — (i — i„)hi+iih. 



b /;3 + {hi — b) }i{! + (/; — /i{) Ifl 



6h 



, >^ 
a^ ^ — 

3 
a = — ] 



Z": 



bIP 



bifl 



i + ibi h 
i + h i 
2 (5 + ^1 h 



2". 



b'- 


+ 


iiil 


+ ^1= 


b'- 


12 
+ 


4 3 ^1 + h' 



12 (2 3 + bi) 



, _ b hr + ^1 hi (h + /<o) 



■2\b h-(b- bi) hi] 
= h — a 



L ^ (<z'3_/3) + ^1 (/3 +^) + ;„(a"3_^3)j 



Determined graphically or 

by experiment. 



Z'= J- 



Z'^J- 



1 [^ (a'3 _/3) + ^1 (/8 + ^ _ ,-3 _i3) + i^ (a'lS -gS)^ 



Determined grapliically or 
by experiment 



Z'= I 

a' 
Z"=.l 



b h + {hi— b)hi-\-{h — /ij J 



bk 



t+h. 



bi h^ + ik^ 



i{i-f)+bi(f+g) 



h{/\e-i-k)-Vb.{a"-gi 



THE CONSTRUCTOR. 

SECTION TKELS^I^Caiitinucd). 



No. 



XVIII. 



t«- & — -^ 



it 


,■,*,.>, 


1 


a° 


1 1 




' 


%i< 




(«— 


— b-- 


— Ji 


\l\~ 


-..> 


1 


-* 


A.J 


"T- 



XIX. 



XX. 




XXI. 



XXII. 




XXIII. 



»..-Jf — It 



l.-*-.J 



^.....-1) fc 



XXIV. 




Moment of Inertia y. 



7 XAJ^i"'"'-^''^ + '' (""-■^'' +h{/'+ <^"^)\ 



Determined graphically or 
by experiment. 






- ^■1= 0,0491 (/* 



-r- (^-1— </i-i) = o.049t{^*— 4-1) 
64 



64 



. bKi 



(Parabolic section.) b k^ = 0.0457^/^3 

175 



XXV. 



XXVI. 




*"'b---* 



FES-. 



XXVII. 



! 4 4'?4-f i 



[» i..!r. ^ r ? i 



,? ! 



» 



-iji— * 



Distance c 



Determined graphically or 
by experiment. 



=_o.5755 rj 
= 0.4244 r 



Section Modulus 'Z. 



Z' = 


J 




a' 


Z"- 


J 




a" 


Z' = 


I 




a' 


7.11 - 


J 



^(«'-/)+^V+«") 



Areay^ 



■5 («'-/) +^=(/+^) 
+ ^3 {a" —£) 



32 



32 d 



-im 



Z' ^ o.rp rS 
Z''=^ 0,26 ?3 



a' = I A 



— r^a'< + ^(/<3— rf3) + ^(/j_a')1 
12 L 10 -* 



- \b (a'3 — /3) + i, (/3 _^3 + ,[J _ /3) 4. ^2 (a'/3 _ ^)1 



64 



-r6(a'3-/3)4.^j(/3_^^3) +^„(^3_,-3+ ;3_,„3) 

3 L 

+ ;3(*!_/3)+i^(„"3_X-3)J 



Determined graphically or 
by experiment. 



Determined graphically or 
by experiment. 



Z = — li k- ^= o .m, d k- 

35 

Z" =^ b k- = 0.076 b k~ 

105 



~(d"—di^) 
4 



ik-r, 
4 



Ji*A 



l(o.5S9<l'4 + ^(A3-<i3)+33(/,_^) Z^o _|.,^(/, 

3/:^ ' 4 



y 



Z' — 
Z" = l 



h-d) 



+ b^(a" — k) + -d'^ 
4 



Z" = =L 



i [a -/) + b, IJ-- 


-g) 


+ b« (r 


-i+i- 


,n) 


+ b,{k- 


-i) + h{<^ 


-% 


+ ^(.' 


-ff 





THE CONSTRUCTOR. 



Value of the Quantity S. 

The limit of elasticity in a deflected beam, both on the ten- 
sion and compression sides, will be reached when their respec- 
tive strains S become etjual to the modulus of resistance. It is 
therefore of great importance to select such a value for S, that 
the modulus of resistance may not be reached on either side. 
These conditions will be met for sections which are symmet- 
rical about two axes, hy taking the lesser of the two values of 
S, as in the case of cast iron, the modulus for tension should be 
used. 

In those sections in which a' , differs from a" , the first thing 
to be determined is the position of the tension and compres- 
sion sides. Let 

a = the greatest distance from the neutral axis on the tension 
side. 

<7i = the greatest distance on the compression side, 
T= the modulus for tension. 
7i = the modulus for compression, 
/If = the statical moment of the bending force, 
m = the coefficient of safety, so that for double, triple, 
safety, etc., in = 2, or 3, 
Then we may take : 

a r T J 

When— > — then M = 

(?i 71 1)1 a 

a T Ti J 

When -< — then 3T = 

Ci 71 in ai 

a T TJ Tx J 

When-- = then M= °^ 

a^ 71 m a m a\ 

T I 
Example. For Cast Iron — = — . 
Tx 2 

Taking the parabolic section No. XXXV. CE=g h, ai \ h. 

This gives -^ — ~ \, so that — — ^ ""t^i ^^^ ^^'^ -^f ^"^ ha^'e 



■10,650 10,650 

and M = 0.114 ^« ■ 



"1 



71' 



With wrought iron, in which T^ T^ no investigation is necessar;^^ 



Sections of Uniform Resistance. 

In order to use the material to the greatest advantage to re- 
sist bending strains, it is necessary to pay especial attention to 
its distribution, particularly in those portions which are furthest 
from the neutral axis. The best economy is attained in this 
matter when the section is shaped so that the strains on both 
tension and compression sides shall reach the elastic limit 
simultaneously. 

For this purpose it is necessary to make 

«, T ^^ 
Sections which are thus proportioned are known as Sections of 
Uniform Resistance. Wrought iron sections which are sym- 
metrical about two axes fulfi' these conditions, since 7^= T^. 
For cast iron, when the bending strain is exerted constantly in 
one direction, it is best to make ffj = 2 a, for 7] ^ 2 71 

Taking these conditions into consideration, the following 
sections (Figs, i, 2, 3) have been drawn, in which b and b^ may 
tave any desired proportion to each other : 




FIG. I. 



FIG. 2. 



For these sections, when b^ = b, we have : 

/ = 27Si* 440/;* 992i5* 

Z = 34. 8i' 55 i' 02.45' 

F ^= igb'' 25(5- 40.86^ 

= I 0.97 .04 

The tension side is nearest to the neutral axis. 



section modulus is determined from the value of 



/ 



Since the 
, .S is always 



T 
to be taken as — L. F is the area of the section, and ^ is the 

in 
proportional economy of material, the cross section of Fig. i 
being taken as unity. 

The value of 9) may be determined thus : 



I? = 



/3 






(8) 



in which the sub-numbered letters belong to the required sec- 
tion and the un-numbered letters to the given section whose 
economy is to be taken as unity. In this equation F^ pb'', 
Z = ah^ and .? is taken equal to S^ except when the ratio of a 
to a-x is not the same for both sections. It will be seen from 
an examination of (8) that a slight variation from the exact 
proportions is not very material. When the bending force acts 
alternately in opposite directions, so that the strains are re- 
versed, the sections which are symmetrical about two axes are 
the best for cast iron as wcli as for other materials, and the 
smaller value for 61 should always be taken under such circum- 
stances. If the force is constantly changing its direction, so 
that the neutral axi; passes through the centre of gravity, the 
most economical section is that of a circular ring, its resistance 
being greater than the cruciform or star-shaped sections, such 
as X., XII. and XX 7., Table ? 7, since there is in the former 
case a constant prop jrtion of the section and the greatest dis- 
tance from the plane of the bending. 

Example. A projecting beam of cast iron loaded as in No. I., g 6, carries 
a weight P-- 5,500 pounds at its extremit.v, the length being 78.75 inches. 
Taking the cross section of the shape Fig 2, we have by equation (4): 

.fl/= 5,500 X 78.75, Z=isb'^ 



5,500 X 78.75 = 10.650 X 55*' 

^= V 5,500X78.75 ,^ 0.3V 
• 10,650 X 55 

The sectional area will then be 25 (0.9)' = 20.25 sq. in., as de- 
termined by the constant given for the section Fig. 2. If the 
security be taken at ij, 

21,300 
.5'= = 14,200. 

This gives a lighter beam, and according to equation (8) its 

weight would be (— — )' = 0.825 of the preceding. 

? 10. 

Bodies of Uniform Resist.ance to Bending. 

A body is said to offer uniform resistance to bending when its 
shape is so chosen that in all sections of its length the maxi- 
mum strain, 5, for tension or compression has the same value, 
and the general form of equation (4) for such bodies is 



1}U 
J 



= Constant. 



(9) 



Bodies shaped so as to oppose a uniform resistance to bending 
are frequently used in machine construction, approximations to 
the exact forms being often adopted, examples having already 
been shown in § 4. A variety of such shapes are given in the 
following table. 

The deflection in bodies of uniform resistance is of necessity 
greater than in prismatic bodies of the same strength. In many 
of the examples of the following table the deflection, /, is 
given, and in I. it is double, and in V. iJ- times what it would 
be in prismatic bodies similarly loaded. 

The elastic line for the following bodies, when exactly formed, 
is determined from the following equation : 

d^y __ Mo ao (10) 

dx'^ EJo ax 
in which 

Bio := the moment of the bending force for any given sec- 
tion, 
Jo = its moment of inertia, 
ao = its greatest fibre distance, 

ax = the greatest fibre distance on the same side as ao for 
any other section at a point x . 

For the radius of curvature, p, of the elastic curve at a point 
whose co-ordinates are .r, y, we have : 



P= ^J^a^ 



(") 



71/0 ao 

which value is constant, and represents a circular arc when 
ax = ao ; that is, when the section is of uniform height at all 
points, as in V., X., XIV. 



THE CONSTRUCTOR. 



Ko. 



n. 



HI. 



IV. 



V. 



VI. 



VII 



VIII. 



iX. 



XI. 



Fori 



y^— -X — -a 



„..}. — 



Application 

of 

Load. 



Equation. 



For rectangular section 
- f- _ £ 
b Jfi " I ' 

In Cases I. and II,, 
z~6. 






Parabolic truncated wedge. 



Approximation to Form I,, 
Truncated wedge. 



- -1- •; 




Sustaining 
Power. 



S t Ifi 
61 



_ S /, !fl 

6 I 



61 



Approximation to Form II., 
Truncated wedge. 



Normal wedge. 






Cubic parabolic truncated pyra- 
mid. 



Approximation to Form VI., 
Truncated Pyramid. 



Truncated cone ; approximating 
to the form given by equa- 
tion 

y 



/^ y -I 



For rectangular section, 
z y~ X- 

, y X 

z=-0; rL ^ — 

/: / 

Wedge. 



Parabolic-sided wedge. 



' y _if^ 



Truncated pyramid on semi- 
cubic parabola. 



_ S i !fi 
6 1 



Volume. 



5 i Ifi 



S i Ifi 
6 I 



Sir d~' 

3= ^T' 



S b Ifi 
3l 



S blfi 



S b h"- 
3/ 



i- bhl 
4 



— bid 



J- bhl 
5 



19 bhl 
27 



i9 ^IcT- 
loS 



Lbhl 



I. bhl 
3 



J. bhl 
7 



Remarks. 



Deflection of the free end ; 



Tile elastic curve is a parabola. 



Weaivest section at the bas6. 



In normal conditions the elastic 
curve bisects the angle of thfi 
wedge. 



Elastic curve a circular arc. 



Jo 



: JoE' 
b 7:3 



Equatio: 



y 3/ ^ 

h yi I 



is of great- 



est importance when all the sec- 
tions are similar. 



Weakest section at the base. 



To obtain the same strength as 
in Forms I. to VII., make 



s r i6 b 

y r- /T 



A very useful form when the 
sharp end is removed. 



Elastic curve a circular arc. 

f=}- ^^^ 
aJoE'' 

Jo = 



Applicable to stone brackets, 
etc., in architecture. 



lO 



777^' CONSTRUCTOR. 



No. 



XII. 



XIII. 



XIV. 



XV. 




Application 

of 

Load. 



Equation. 



Approximation to Form XI. 



For rectangular section 

-'■Hi 

Wedge, on semi-cubic parabola. 



1„ 3/"" 
' f 1 



Sides on cubic parabola. 



z 


b 


y 


~ h 


y 

h 


X 

~1 


Py 


ramid. 



Sustaining 
Power. 



5 /' m 



S li h- 



S i /i' 



Si A2 



Volume, 



1^ bill 
27 



i-ihl 
5 



Lthl 



i bkl 
3 



Remarks. 



Weakest section at the base. 



Fundamental" shape for archi- 
tecture. 



Elastic curvea circular arc. 



h = 



bh^ 



Value depends upon the sim- 
plicity of the form. 



The preceding are only a few of the simpler forms which 
may be used, and it would be easy to multiply examples. 

By altering the breadth, or height, the relations become 
more or less complicated, as the case may be. 

For instance, in Case I., which is based on the parabola 

z rx~ ^' 4/'r 

r-= «/ i, it may be made the biquadratic parabola, —^1/ ' ' 

etc. Combination sections give rise to new forms, and a great 
number of combinations may be made. Bxamples will be 
found in the chapter on axles and shafts. 

The following discussion of springs will also give some in- 
stances of special forms, in which the neutral surface is irregular. 



Resistance to Shearing in the Neutral Plane. 

Since in a deflected beam there is on the tension side a con- 
tinual tension, and on the compression side a continual com- 
pression of the respective fibres, it follows that the neutral 
plane is subjected to a shearing action, and this must not be 
neglected in determining the width of the beam. 

The lower limit permissible is indeed a matter not likely to 
be reached, but at the same time it should be investigated. 

Calling the least permissible width Zo, and the mean force on 
either side of a given section R, then in order that the shearing 
strain at the neutral plane shall not exceed a value So, we must 
have : 

>^ U ■ (14) 

''So 2/ 

in which So should in no case exceed f of the lesser modulus 
of resistance of the material under consideration (see ? 5). J, 
as before, is the moment of inertia of the section, i.e., the 
summation of the products of the elements of the section by 
the square of their distances from the neutral plane, while 1/ is 
the statical moment of the section, i.e., the summation of the 
products of the elements of the section by their distances from 
the neutral plane. 

For the rectangular section No. I., Table (? 7), 



'^0 =-=- 



U=' 

4 ■ 
and for the double T section. No. VIII., 

l_j_ b h■'■ — {b — b^) /n^ 

4 
R is to be chosen according to the case under consideration, 
as, for example, in No. II. Table (§ 6) for all sections between B 

p 
and C, it is equal to the reaction — , etc. 

2 



Equation (14) is not so much used to determine a value for 
Zo, as to find out in any case whether the breadth of the neu- 
tral plane has been taken too small. As a matter of fact, this 
is a question which very seldom arises in ordinary construc- 
tions, especially in machine construction. 

4 

If in (14) we give zo any desired value, and make So ^ — s 

we obtain 

4 Zo 2/ 

and substituting this in equation (4) we get : 

M^5 U (15) 

R % zoa 

M ■ 

— -^ is the lever arm of the force R ; this we may call ^. 

U : Zo a contains one of the height dimensions of the section ; 
hence equation (15) expresses a relation between two dimen- 
sions of the body under consideration. For a simple rectangu- 
lar cross section, taking the value of U, given above, in which 
h , h 16 

A greater value for h must not be taken if we do not wish the 
shearing strain to exceed the extension or compression in the 
tension and compression sides of the beam. These considera- 
tions are often of importance for the danger section, as, for ex- 
ample, in No. II., Table {\ 6) for the point B. In this case 
/ / = 8 

A = — and we make — ^— . This limit of height, however, 
2 . A 5 

is so great that it is very rarely reached in practice. 

The most important application of this principle is found in 
the case of notched beams of wood, such as often occur in 
building construction. In such cases the resistance of the neu- 
tral plane is often ven,' much reduced by the cutting of the 
notches, sometimes to one-half what it would be in the solid 

beam, and making a corresponding reduction in the value of 7- 
For the double T section we have : 
h 16 



Zo = b, and a = . 



[f-a-oay] 



If the brackets in 
fraction the value of - 



the denominator contain an improper 
— will approach the upper limit, but lor 



THE CONSTRUCTOR. 



II 



all ordinary cases this value is very great. The nearest ap- 
proach to this shearing action probably occurs iu T beams 
where the flange joins the web, but examples are very rare. 



BEAMS WITH A Common Load. 

When two prismatic beams are united iu the middle, and at 
that point subjected to a force P, the beams being supported at 
the ends, they will both be deflected, and the sum of their re- 
actions P' , and P" , enter into the support of P. 

The double reactions are found from the formula in Table 
(I 6), No. II., column 2, as follows : 

P' _ J' E' l"^ 
p77 — J" E" 17^ 
and since 

S"J'' 



P' = A 



J"E' 

S' J ' 
a' I' 



and P' 



I" 



we get 



S' 



'" ~ E" a"\l' ) ^ ' 

If the two beams are of the same material (E' = E'^), to ob- 

tain equal security, the product-^ ( \ 

If the beams are not the same length then a' = a", i.e., the 
heights must be the same unless the breadths are equal to each 
«ther. 




"*•--.. 



PIG 4. 

Kxample. A cast 5ron support shaped like a cross. Fig. 4, must support a 
weight, P, at the intersection. The lengths of the arms are to each other 
as 3 : 2. In order to obtain equal security in the four arms, which are of 
prismatic shape, we have from (16) 

Hence the cross section of the short arms must be to that of the long arms 
as 4 ; 9, and if the arms were of the same section the supporting power of 
the short arms would be to that of the loug arms as 9 ; 4. 

It also follows from the preceding, that rectangular sheet 
metal plates carrying a uniformly distributed load are stronger 
parallel to their shorter axis than parallel to the longer axis. 

For given loads and materials formula (16) may be used to 
govern the choice of dimensions and the relations of length to 
breadth. 

For beams of cast or wrought iron resting upon each other, 
a suitable proportion may be secured by taking the sum of their 
several supporting powers as the supporting power of the 
combination. This is often a matter for consideration in 
strengthening existing structures. 



Resistance to Torsion. 
Resisting Power and Angle of Rotation. 
A prismatic body which is subjected to the action of a force 
couple tending to rotate it about its geometric axis, opposes to 
such action its Resistance to Torsion. Under these conditions 
the elements in a normal section are subjected to a shearing 
strain, and until the elastic limit is reached there exists an 
equilibrium between the external rotating forces on the one 
hand and the strain moments of the various elements of the 
section on the other hand ; both being taken with regard to the 
polar axis of the centre of gravity of the section, i. e., the axis 
passing through the centre of gravity of the section and at 
right angles to it. Resistance to torsion may properly be con- 
sidered a higher species of resistance to shearing, to which it 
bears the same relation that resistance to bending holds to ten- 
sile and compressive strength. 
Let: 
■ M = the statical moment for any given section of the 

rotating force, 
Jp = the polar moment of inertia of the section, /. e., its 
moment of inertia taken with regard to its polar 
axis (see \ 14), 



a ^ the distance of the farthest elements of the section 

from the centre of gravity, 
.S =: the shearing strain in the elements at a distance a, then 

a 

Jp 



(17) 



If the body is of a uniform section, then =^ is constant. Now 



(iS) 



if A be the lever arm of the rotating force P, for a moment M, 
the weakest or danger section will be that for which Af is a 
maximum, and for it we have 

/'=_£. J± 

Am a 

in which Am, is that value of A, which gives JIf, a maximum. 
The limit of elasticity is reached, as in the case of shearing 

action, when .S ^ — of the lesser of the moduli of resistance 

5 . 
for tension or compression (see § 5). This is plainly visible by 
a comparison between the action of bending and twisting. 

The relative rotation which two sections of a prism at a given 
distance apart make with each other is called the angle of tor- 
sion. It is represented by the letter 1? ; and for two sections 
separated by a distance ^, we have in general terms : 

J^ = 7^ (19) 

a.v Jp G 

in which G is the modulus of torsion for the material used, and 
is equal to — of the modulus of elasticity E. 

In the following table will be found the values for : 
The moment M, at a given point x, of the prism, 
The force P, according to formula (18), and 
The torsional deflection in terms of angular measure, or in 
other words, the angle of torsion 1?. 
These quantities are given for a variety of cases, as shown in 
the cuts, and from them total moment, PR, of the twisting 
force may be determined. In case IV., .Sis the point of appli- 
cation at which the collected forces, with a lever arm R, would 
act, if concentrated to produce an equivalent result to the sum 
of the separate efforts, lo being the distance of the point S from 
the immovable end of the prism. 

Questions relating to torsion are of varying importance in 
machine construction, and come especially into consideration in 
calculations relating to springs. Case IV. illustrates the condi- 
tions which occur in determination of mill shafting. Cases V. 
and VI. occur in machine framing. 

?I4. 
P0I.AR Moment of Inertia and Section Modueus. 

The polar moment of inertia, Jp, is easily determined, since 
we have 

//=/i + /= (20) 

in which _/, and J„ are the equatorial moments of inertia taken 
with regard to two axes at right angles to each other, and whose 
values are given for a variety of sections in the table of (§ 7). 

From this may be obtained the polar section modulus — =^ Zp 

for use in the preceding cases. An exception must be made for 
those sections in which we have not _/, ^ J„, as in cases III., 
VII., XII., XX., XXV., etc., I 7. For these it will be necessary 



to make a special correction in the values of Jp and 



Jp 



■■ Zp, to 



provide for the warped surface which is assumed by the section 
under a heavy torsional strain. 

For a rectangle, which is a section of frequent occurrence in 

machine design, the corrected value of Jp and .2}* = — is given 

in the following table, while for the circle and the square no 
corrections are necessary for the values obtained from equa- 
tion 20. 

Example. A cylindrical prism of wrought iron is subjected to a torsional 
strain applied as in case I. of the following table. The force P ^^ 1,000 lbs., 
and the lever arm R = 24" ; while the bar is 4" in diameter and 4S" long. 

These quantities give for S, the strain at the circumference 



Jp 

16 



16 PP 
1,000 X 24 



1,909 lbs. 



3.1416 64 

and to get the angle of torsion we substitute this value in the formula : 

i.qcq 48 

= ' r ^ ■ ^^ = 0.004 

11,360,000 2 

whicli corresponds to an angle of about o° 14'. 



12 



THE CONSTRUCTOR. 



No. 



II. 



III. 



IV. 



V. 



VI. 



Application, 



•I 

it*' 

4 




*^ ' 















^ 



T 



p 




•..;y 



L: 



ffffff^tf^': 






i3 



Moment j1/. 



M = P R for all 
points between A, 
andB, 



M=PR :L^ 

r- 

P I! = the collected 
moment of all the 
twisting forces. 



Jtf= the sum of the 
moments within jr. 



In the portion c: 

M=P R^l 
I 

In the portion C\ : 
M=P R L 



M=P 



^(7-t) 



Twisting Force P, 



P = 



SJp 



_SJp 



SJp 
aR 



p^SJp 
aR 



When C\<^c 
p^SJp I 
aR c 



P— 



aR 



Angle of Torsion i9. 



iS = 



PRl 
Jp G 

_s_i 

~ G a 



2 Jp G 
2 G a 



i Jv G 
3 Ga 



■ JpG 

_SJo 

' G a 



PR ccj 
"jp G I 

G I 



tJ 




I PRl 




i JpG 






I 5- / 




^ 


iG a 



Remarks. 



All sections between A^ and 
B, are equally strong. 



Weakest section at B. 



Twisting forces decrease uni- 
formly from 5, to ,<4. Weak- 
est section at B, 



General form of Cases I., II. 
and III. Weakest section 
at B. The value of ^ in 
III. will be reached in IV., 
when /o = — • 
3 



The shorter portion cx is the 
weaker. 



Weakest points at A\ and B. 



If we wish to reduce d, so that ^ shall be equal to one-half the modulus of 
resistance for torsion, i. e., = — ■ -— - • 21,300 = 8,520 lbs., we make 



P/$ ■■ 



d^ ^^ 
ttS 
or about aj^ inches. 
In this case the angle of torsion would be 

11,360,000 1.2; 

which gives an angle of about i° 39". 



-,^ 16 X 1,000 X 24 = 2,^2'/ 
3.1416X 8,520 



= 0.02S8" 



SECTION TABLE. 



No. 


Section. 


Polar Moment of 
Inertia yij. 


Polar Section Modulus, 

, Jp 
Zp=.^ 


I. 


#} 


32 


16 



No. 



II. 



III. 





■—i-v 



Polar Moment of 
lnertia_/^j. 



6 






Polar Section Modulus, 
7 _ Jp 



33 
3 n/ : 



3\/ i°' + hi 
Approximately 

3 (0.4^5 + 0.96A) 



THE CONSTRUCTOR. 



13 



I 15. 

Bodies of Uniform Resistance to Torsion. 

In order to make a body of uniform resistance to torsion it is 
necessary to take such sectional areas at various points as shall 
make in equation (17), So. constant, and also to take 

— ? =: constant. (21) 

Jp 
In case I. of the table in \ 13, for all sections M ■= PR, and 
hence in this case the body should be prismatic in shape. For 
cases II. and III. the necessary formula are given in the follow- 
ing table. For such bodies the angle of torsion is greater than 
for those of prismatic shape. The angle for each is given in 
the table, and is derived from the following : 

i± = ^L (22) 

in which y^ is the polar moment of inertia for the section taken 
at the point x. 



Form. 



tion. Equation and Angle of Torsion. 




Circular section 

y 



:^: 



16 

a SI 

G a 
Appro.\imate form = a trun- 
cated cone, with extremity 

= — d. 



Circular section 



,? = 6. 



S— d^; 

16 

/ 



Approximate form = .a trun- 
cated cone, with extremity 
d 



For other bodies of uniform resistance to torsion, see Torsion 
Springs (§ 20). 

?i6. 

Resistance to Bucki<ing. 

Combined Bending and Compressive Strains. 

A prismatic body is subjected to combined bending and com- 
pressive stresses, to which it yields by buckling, when its di- 
ameter is comparatively small in comparison with its length- 
Under these conditions a compression applied in the direction 
of the axis is opposed, both by the resistance of the body to 
compression and also to bending, with this difference, that in 
this case the lever arm of the bending force is not the abscissa, 
but the ordinate of the elastic curve. From this it follows that 
(neglecting some very small elements) any compressive force 
P, capable of producing a bending, would do so even up to the 
breaking point, provided that the laws of perfect elasticity held 
good until rupture occurred. This would only be true if the 
theoretical resistance and the breaking load were the same, and 
the elasticity of the prism held them in equilibrium until the 
final yielding of the point of application of the force P oc- 
curred. 

In the following table (p. 14) the principal formulae are given 
for a number of the most important applications of these buck- 
ling stresses. In the table 

E = the modulus of elasticity of the tnaterial assumed to be 
of prismatic shape ; 

J = the least moment of inertia of its section taken with ref- 
erence to a line of gravity, for example, in a rectangle of 
which the greater side is b and the lesser side h, according to I 7, 
h b^ 

12 
It may be remarked that the valuable experimental researches 
of Hodgkinson, as given in his rules, show a somewhat smaller 
breaking load than the formulae in the table ; this, however, 



does not detract from the value of the latter, since these are only 
strictly correct for perfectly elastic bodies, but at the same 
time they will be found practically reliable if the force i°is not 
permitted to exceed a definite proportion of the breaking load. 
Different materials demand a different factor of safety. For 
cast iron, % to Yb the breaking load, or less, and for wrought 
iron the same, and for wood \ to ^\j, or jV, should be the limit. 
These inequalities often arise from the fact that it is not 
always easy to determine which of the applications of the 
table really meets the case in question. In order to determine 
the actual security from rupture, it is often necessary to make a 
comparison with other existing strains. From this standpoint 
the ratios of diameter to length in the following table have 
been determined in order that the resistance to compression and 
to buckling may be as nearly alike as possible. 

In Hodgkinson's experiments it was shown that columns 
standing upon flat bases were nearly as strong as those which 
were firmly fixed at one end. 

In the third section many applications of these formulae will 
be given. 

? 17- 
Coi<UMNS OF Uniform Resistance. 

Columns subjected to combined compressive and buckling 
stresses are said to be of uniform resistance when its various 
sections are so proportioned that a very small degree of buck- 
ling will produce the same strain in each section. 

For case II. of the preceding table, when the section is circu- 
lar, the following formulse (by Redtenbacher) may be used : 



This may be separated into a double equation by making : 



y 

h 



which gives : 



(4) 



=: (2 ^ — sin 2 I 



(23) 




'•c>7^i 



From these equations a limiting 
curve may readily be found, whose 
abscissas are those of a cycloid, and 
whose ordinates are those of a 
sinoide, and which may be called a 
cycloidal sinoide. A method of 
drawing this curve is given here- 
after, in the discussion of connect- 
ing rods, and the approximate shape 
is also shown in the second form of 
Fig. 5, in which the outline is a 
circular curve, or at least a line of 
very slight curvature. The strength 
of these columns ma}' be taken as 
34 that of a cylindrical column of a 
diameter h and length /. 

I 18. 
Compound Stresses. 



fig. 5. 



It very often occurs that a variety 
of forces act upon a body at the 
same time and in a variety of ways, so that, for instance, a sec- 
tion is subjected at the same time to tension and bending, or to 
torsion and bending, etc. 

The resistance and the maximum strains are then to be de- 
termined in a different manner, according to circumstances. 

In the following table ip. 15) are given the principal formulsi 
for some of the more commonly occurring cases. 

Let: 

^ = the greatest strain at the weakest section ; 

Z = the section modulus at the weakest section, which 

latter is indicated at B in the figures ; 
F ^ the area of the section ; 
J ^ its moment of inertia (2 7) ; 
Mb = a bending moment ; 
Md = a twisting moment ; 
Jlf; = an ideal moment, so that 
(Mb)t = an ideal bending moment, and 
{Md)i = an ideal twisting moment. 

An examination of these formulae will show that in many 
cases the combination of strains is a matter of importance. 



H 



THE CONSTRUCTOR. 

BUCKLING STRAINS. 





Application. 


Theoretical Support- 
ing Power. 


Remarks. 


The column is to be considered under compression, when : 


No 


For circular section 
-^- is less than 


For rectangular section 

—r- ib = lesser side). 



Material. 






II' 






5 


sV* 


Cast iron. 


I. 




A 

* 


p ^- JE 

4 r- 


Freely loaded post. The 
lower end fixed. Weak- 
est section at the point 
of attachment. 


6 


8 


Wrought iron. 
Wood. 




^., j,^ ,„,, 








i; 
















/ 








lo 


iiK 


Cast iron 


II. 




\ 


;J 

-1 


/2 


Post free at both ends, 
but kept in the line of 
the axis. Weakest sec- 
tion in the middle. 


24 


28 

'3J< 


Wrought iron. 
Wood. 














.A' 
















i 
1 








14 


16 


Cast iron. 


III. 




i| 




/3 


Post fixed below, and 
held in the line of the 
axis above. 


33 


33 


Wrought iron. 












l6 


19 


Wood. 




f 


'' 






























,.,, 1/-:^ 














1^ 










\ 






20 


23 


Cast iron. 


IV. 




t: 




^ /2 


Post fixed at both ends 
in the line of the axis. 
Weak points at the 
ends and in the mid- 
dle. 


48 


56 


Wrought iron. 






i ' 






23 


27 


Wood. 




t„ 


.- 











For example, in case I., if j*? ^ — ,''■£■, if the load is hung at 

2 



tlie edge of the section, P - 



Sbh 



, and hence is only one-fourth as 



great as it would be if applied centrally. If th^ section is circular 



(d), we have P = 



S~d^ 
4 



J? 



making Jf ■- 



P=—— d\ 

5 4 



and the sustaining power is still less than with a rectangular 
section. Case III. is derived from I. and II., and may be 
changed into either by making a, ox R ^ o. 

The so-called ideal moments are especially useful in these 
calculations. It will be noticed that in the case of elliptical 
and rectangular sections, li is taken in the plane of bending. 
These dimensions being known in advance, since the choice of 
profile is frequently permitted, it is possible by the use of the 
ideal moments to consider the question of combined strains, 
eince the quantity in the parenthesis to the right is the expres- 



sion for the lever arm of the force P for each case. This can 
generally be readilj' determined graphically, and so determined 
just like any case of ordinary bending. 

For example, in case II., if o^ 45°, we have cos a = sin a = 
0.707 for the value of /i, and the section at B is to be calculated 
as if acted upon by a force P, with a lever arm o. 707 / (the pro- 
jection of / on the plane of attachment) + 0.707 — 

6 

In case I., making R = o, for a circular section {3Tii)i = P ~- 

o 

and substituting 6" — d^, we get P^= S — rf^, as we should since 

the stress is now purely tensile. In this case — - is the lever arm 

8 

■which, if acting with a bending force P, would produce a strain 
of the same amount as that in the line of the axis. This is 
only rigidly exact when the shearing action which occurs in 
bending is neglected. Many useful applications of cases IV. 
and V. are found in discussing axles and shafts. 



III. 



IV. 



THE CONSTRUCTOR. 

COMPOUND STRAINS. 



IS 



Application. 





't:l^\ 



B/ 




Sustaining Power. 



I +yf ■ 



for rectangular section, {bJt) 
Sbh 



P= ■ 



1+6- 



for rectangular sectionj {bh) 
Sbh 



for rectangular section, (Bh) 
Sbh 



I R 

cos a -\- d —r-{sin a H — —cos a) 



PI is a bending moment Mb, 
PR is a twisting moment Md. 



SZ 



Mfflill 



\/ Ml- + il/o- -\- "2 Ml Mn COS a 

in which Mi is the bendiJig moment 
of Pu Mo that of P2. 



Ideal Moments. 



Ideal bending moment for stress 5: [Mi, \i = P [R -\ j. 

For rectangular section {Ih) : 



For circular section (d) 
d 



Mb 



>=-(-+4) 



For elliptical section {bh) : 



(-0'=-(^+4) 



Ideal bending moment for stress S: ( Mb 



For circular section (d) : 

(Mb ]i =P I i sin a + 

~^cosa^ 



For elliptical section {l>h) : 
(Mby^P 1 1 sin a + 
-^cosa'j 



For rectangular section {bh) : 

I Mb )i —P (l'"' « + 

-^cos.'j 



Ideal Ijending moment: [Mb ]i = P (R cos a + I sin a + - 



For circular section {d) : 

yMb \i = P (r cos a. + 

i S2n a -i — cos a 1 



For elliptical section (S/i) : 

{Alb ji = P [R cos a. + 

I sin a H — cos a. 1 



For reotangular section {bh) ; 
I Mb \i = pIr cos a + 



Ideal bending moment for the stress 6" : 

(^Mb y --^Mb + 

Ideal twisting moment ; 



V 
V 



'V + ^'i 



-Mb+ . 7lV + M^' 



rdeal bending moment for the stress S: 



(Mb ) i = I /"^l/r + -^^2- + 2 ^1-4 ^l^a ^-^-s- a 

In cases IV. and V. it is supposed that the section is arranged in four symmetrical 

portions about two lines of gravity, perpendicular to each other. 



i 19- 

Resistance of Walls of Vessels. 

Boilers, Cylinders, Etc. 

The fallowing table will serve to determine the resistance of 
the walls of cj'lindrical vessels subjected to pressure for the 
cases which usually occur in practice. 

The theory of resistance under these conditions is not fully 
settled, especially in the case of comparatively thin shells sub- 
jected to external pressure, for which the corresponding formulae 
do not give satisfactory results. 
In the following cases, let : 
p = the unbalance"d pressure upon the walls of the vessel ; 
.S = the maximum stress for the material used ; 
Ji = the modulus of elasticity of the material : 
r = the radius of the vessel ; 
(5 = the thickness of the walls. 
Although only approximate, the formulse for cases I. and II. 
hold good up to the limit of rupture. 

Examples: i. Given a wrought irou cyliuder, 40 inches diameter, %" 
thick, with a stress upon the material of 11,500 lbs. Under conditions of 
case I., the internal pressure permissible would be 



P—11. 



2. A spherical vessel of the diameter and thickness griven above, according 
to case II., ■would have a safe resistance 

P= 23,000 — — ^ = 431 lbs. 
20 

3. A plate held as in case IV., 40" dia., ^" thick, and a pressure of 212 lbs., 
with a maximum stress 5" ^ 11,500, would have a thickness 



Y 3 y 11,500 



' 20 X 0.S16 X 0.136 ^= 2.22" 



or about 23<( inches. 



The deflection 7^ which a circular plate gives under a force p, 
may be determined, according to Grashof, by the formula for 
case III. : 



and for case IV. : 



7_ 
(J 



A. 
E 



(24) 



(25) 



= 11,500 X 0.0185 = 212 lbs. 



Example : The plate of Example 3 preceding, with a value of £ = 28,400,- 
000, would have a deflection of 



\2.25/ 



Y 2" 



i6 



THE CONSTRUCTOR. 



RESISTANCE TO PRESSURE. 



No. 



> 



Application. 






Pressure p. 



/=s(/".+^-. 



p = S{ 



■(4-r 



^=^(-^) 



25 



{■ 



--r^f 



For vessels wiiose walls are required to be made very thick, 
as in the case of the cylinders and pumps of hydraulic presses 
or for cannon, etc., the preceding formula do not apply. Under 
these conditions the relative radial distances of the various por- 
tions of the thickness of the metal vary greatly, and their 
relation has an important influence upon the resistance. It 
is the relation which exists between the various stresses at 
different points which governs the various formula for the 
thickness of the walls, which are given below. Brix calcul^es 
the stresses at different points on the radius upon the supposi- 
tion that the internal diameter is not altered by the pressure ; 
Barlow admits such an alteration by pressure that the area of the 
annular section of metal is not reduced ; Lam6 makes neither 
of these assumptions, but calculates very closely the changes 
in the various stresses which are caused by the internal pressure 
at each point, and in this way has obtained the most reliable data 
as to the real behavior of the particles of the material, accord- 
ing to the modern theory. The results of the three theories are 
given in the following table : 



Quantities. 









/ = 






P = 



^lognat eS — i 



25-- 

r 

2 S 



Barlow. 



5 


'V-' 


S-p 


iS 


p ^ 



2 S — p 



Lame.' 



{r + ^Y + >-■' 



i: 



S+p 
S-p' 



(r 4- 6f — 1^ 



II HS+P) 

2S — P 



For the stress 5' in an annular ring lying between the radii 
r' and r, Lame gives 



[-(^•)']-^[-(,0'] 



If r' is the external radius of the vessel, so that r' = r -^ S, 
we have : 



or if we put ( i + - 
S' 



S /m' + I p //' — I 

2 /i'^ 2 ji^ 



Example : If 5 = ^, that is, /x = 2, S' = —5— ^ ^— p, and as in the px^ 



ceding formulee, taking/ = 



iT, we have 



.y^- 



8 40 5 

This shows that the material is not used in an economical manner in ves- 
sels witli excessively thick walls. 

All three theories admit that the inner portion of the wall is 
strained the most, and hence it is for the inner wall that 5 
should be chosen. The formula; of LamiS, as well as those of 
Barlow, show that beyond certain limits an increase in the 
thickness is not attended with any increase of strength. With, 
a given resisting power S, this limit will be reached when p^Si 
the theoretical resistance will be attained when p = the modu- 
lus of resistance of the material. At this point the internal 
pressure begins to stretch the inner fibres of the walls, and any 
increase in strain will cause rupture. The theoretical limit in. 
this case is reached when p ^ T, which is 

For Cast Iron = 10,650 lbs. 

" Wrought Iron =: 21,300 " 
" Cast Steel = 36,000 " 

Lack of homogeneity in the material may cause the danger 
pressure to be reached far within these limits, the material 
breaking without previously stretching. 

Since stresses exceeding 36,000 pounds are reached in guns of 
large calibre, it is evident that ordinary bronze is unsuitable for 
such conditions, and even homogeneous steel is often unequal 
to the pressure. The erosion of the chamber in the case of 
ordinary bronze cannon also acts to weaken the inner ring of 
material, and must be considered as a chemical deteriorating 
action. 

Various methods have been devised for strengthening guns 
by giving the various layers different tensions. Of these 
methods the principal is that of hooping. The principal result 
of this construction is to produce a compression in the inner 
layer. The pressure of the gases of explosion must then first 
overcome this compression and restore the normal condition 
before it can produce any extension of the fibres, and as a result 
a much higher degree of resistance is secured than when the 
metal is left in its normal condition. 

The calculations of the resistance of hooped guns offer many 
diificulties. If we have not only the inner pressure, but alsc» 
the outer pressure, p' , to consider, we may take the following 
formula, after Lame : 

,1-^-^, = , ^+.fi .. (.7> 



(^+4r= 



S—P+2P' 



Putting I -| =: /i, as before, and solving with regard to p. 



we have : 



p = S 



II- 



+ 2/'; 



/Z^ -f I ^ /i'^ -f I 

in which 6' will become less with regard to/, the greater))' be- 
comes. 

In the case of hooped guns 
/' is not constant and iuvari- 
ble, but depends upon the 
effect which the internal pres- 
sure /> has through the walls 
upon the hoops. 

Referring to Fig. 6, let it 
first be considered that under 
normal conditions the inner 
ring is under no strain, that 
is, J* = o, and also S\ = S"^ 

Now when the inner pres- 
sure j* becomes sensible while 
the external pressure/" = o, 
or at least may be neglected, 
then the layer at r' will be- 
come extended, and the ten- 
sions will be 5/ = S/. The 
stress S/ in the inner side of 
the hoop reacts with a pres- 




THE CONSTRUCTOR. 



17 



sure/', aud substituting this in Lame's formula, making i ^ — -^ 
= //', will give 

Making S' = S/ = S/ and substituting this value of/' in (28), 
gives 



m'+ I 



H"+ Ifi'+l 



According to (26), S' is dependent upou p aud 5, and by sub- 
stituting and transforming, we get 



/=5 



'+1 



. sii±±mM=^ ,30, 



(■ + t)"(- + ^)' + - 



In this case the stress S upon the inner ring is always greater 
than p, but the ratio approaches much nearer to unity than 
before, as the following table shows : 



IWhen 


We have 




And also 




r 


1^ 


Z' 


/*' 


P 
S 


5 
P 


p 


5' 
5 


I 





2 


1 


0.600 


1.667 


0.667 


0.400 


I 


0-5 


2 


1-5 


o.Soo 


1.250 


0.406 


0.325 


2 


I 


3 


2 


0.905 


1.05.7 


0.143 


0-135 



It will be seen that the mere hooping of a gun with a ring of 
the same material as the inner tube adds very materially to its 
strength. If, however, the ring is forced on in any manner so 
as to produce an initial strain p' upon the tube, a still greater 
advantage will be the result. 

If we insert the value oi p' from (29) into formula (28), we have 



/ = 5 



+ 2 5, 



, /<" — I 



(31) 



In this formula S.,' is partially a function of p, and also de- 
pends partly upon the extent to which the tube reacts. This 
latter condition exerts a most important influence upon the 
strength, as we shall see hereafter. 

If we assume that the hoop is under such an initial strain 
that, for the maximum value of />, the value of S^' = S (which 
is doubtless the most desirable condition), we shall then obtain 
from (3 1) 



If (5 = r 
this gives 



<!' = cS, = — r', then we have ;/ ^ 2, /i' =: 



and 



A__3_,i2_ J__.79 
•S' 5 13 ■ 5 65 

This shows S to be less in value than p, or in other words, it 
is possible to permit the internal pressure/ to exceed the mod- 
ulus of resistance without overstraining the material. It is also 
evident that by encircliug the hoops with additional hoops, this 
principle may be extended still further, and the ratio between 
p and 5 still further increased. 

If the material of the gun be taken as ordinary cast steel, 
with a modulus of resistance of 36,000 lbs., the pressure of the 
gases of explosion could not be permitted to exceed 43,000 lbs., 
without causing a permanent deformation of the bore. Recent 
experiments, however, have shown somewhat greater figures 
than the above. 

Some of the later tests in England 
have shown pressures of 25.8 tons on 
the square inch, although this pres- 
sure is considered by some engineers 
to be rather too high to be safe. It 
is quite possible that in this case the 
modulus of resistance of the material 
exceeded that given above ; or the 
interior tube may have been hard- 
ened, which, if properly done, is de- 
cidedly advantageous. 

The compression exerted upon a 
cylindrical tube by an external pres- 
sure, as in Fig. 7, may be determined by an application of for- 




-P' 



mula (27). If we assume the internal pressure/, to be =o,ox 
at least so small as to be neglected, we get : 

(s-f 2/0m^=-s- 

from which : Si — /i^ 



S=—p' 



2 li' 



1 + 



.//. 



fi + -V-: 



(33) 



The minus sign indicates the change from tension to com- 
pression. When the internal pressure = 0, the stress in the ex- 
ternal iibres is : 

2 iJ.' 



which gives ; 



S' = —p'- 



= -/' 






(34) 



This value is less than the preceding ; for by division we ob- 
tain the ratio : 

S' I ^^-f I I (i+r)'+' , , 

— ~ (35) 



5 



{^+ir 



which can only = i, when <5 = o. Hence for external pressure 
the greatest stress is always on the innermost iibres. 

If, for example, S = r, aud hence /^ = 2, then the stress in the 

8 5 

inner wall of the tube will be 5= — — /' and S' ^ — — p' , so 

5 3 3 

that S' = -^ 5. This is a greater proportion than when the 

pressure is from within, as under these circumstances according 

to formula (30) 6" =: — /, only. 

It is not uncommon in machine constfuction to strengthen 
hubs and other parts of machinery by forcing on hoops or rings, 
and the calculations relating to such construction are closely 
allied to the preceding. The following case will serve to illus- 
trate. 

B B- 




FIG. 8. 

In Fig. 8 is a ring B, which is to be forced on to the cylindri- 
cal shaft A. The following applies either to shrinking, or to 
cold forcing. Before the operation the radius of the shaft is >-j, 
and the radius of the hole in the ring r^, while afterwards they 
both have the same radius r. 

Under these conditions the shaft B will be subjected to a uni- 
formly distributed compression 6",, while the inner surface of the 
ring will be under a similar tension S.,- Taking the correspond- 
ing moduli of elasticity E^, and £^, we have from formula (2) ; 

Adding these together, we get : 
_ 5, 5, 



£. 



"- E, 



It is most important for the designer to know the best values 
for 7\ and r,. 

If we call T^ = — =-, we have 



^ _ il :^ I :^ _ E^ E^ 

^1 



(36) 



i8 



THE CONSTRUCTOR. 



5i and 5, are dependent upon each other, and their relation 
is expressed by Lamii's formula : 



S^ = S., 



P-- 



= 5, 



+ 7J-^ 



i + J+i 



which may be abbreviated bj' putting 
5, = 5, p 



This gives : 



^■. 



El n E.^ 



E% El 



(37) 



The difference between the value of the denomiuat6r and 
unity is so slight that in practice it may be neglected, and for a 
practical and useful formula we have 

5, 



■Si , 

V' = IF + c- 
El p E., 



E., El 



(38) 



In this formula we have for the following : 



= 0.5 0.6 0.7 o.S i.o 1.5 2.0 3.0 

T 

p = 0.385 O.43S O.4S6 O.52S 0.600 0.724 0.800 0.882 

We also have from equation (38): 

V' ^1 „„., c _ V- E^_ 

(39) 



Si = 



I + 



E, 



and S„ = 



1 + 



E^ P 
El 



This value of i' is generall}' so small that great care is neces- 
sary, in turning and boring, to secure the correct sizes for r^ 
and ?'2- 

Example : With a wrought iron shaft and a cast iron hub we have 

^l = 28,400,000 ; £0 =- 14,200,000. 
If 6 ^ 2r, then p = 0.8 by the table above ; and we may also assume that the 
stress, S-_>. in the interior of the ring due to the forcing should not exceed 7200 Ibs.^ 
This gives from equation (38) 



'/' = 



7200 



+ 



7200 X o 8 



14,200,000 28,400,000 1408 



.00071, 



or, in other words, the increased diameter of the shaft over that of the hole must be 
0.00071 times its own diameter. 



If we make i/f = 



-, we shall have a stress in the ring of 



1 + 



— X 14,200,000 
14,200,000 X 0.8 



: 16,950, 



20,400,000 

or nearly 17,000 pounds, which would be too great for the ring to stand. 



?20. 

The Cai<cui,ation of Springs. 

The materials used in machine construction are all more or 
less elastic and 3-ielding, so that it is only by a judicious dis- 
position and proportioning that we are able to avoid an injuri- 
ous deformation of their parts when subjected to the action of 
external forces. Indeed, it is the principal aim of the construc- 
tive engineer to keep the various forms of distortion, such as 
extension, compression, bending and twisting, within as narrow 
limits as possible. In the case of springs, however, it is sought 
to utilize this property of elasticity for a variety of purposes ; 
such as to modify shocks, as in the case of buffers and car 
springs, or as a source of motive power in clocks and watches ; 
or in cushions, mattresses, etc. 

All bodies which will permit great alterations of form within 
the elastic limit may properly come under the designation of 
springs. 

The only substances which are of service for springs under 
the action of tension and compression are those which are soft 
and readily compressible, such as rubber ; while the more rigid 
materials, such as wood and the metals, are used in flexure, or 
in torsion. 

In the following table is given a number of forms of the most 
usual springs, both for bending and torsion, with their respec- 
tive properties. 

Next to elasticit}', the property of a spring to be considered 
is the economy of material, both on account of cost and space 
occupied. lu order to make it possible to compare different 
springs in this respect, the relative volume is given in the last 
column of the table, for the same load and application in the 
different cases, the volume of the triangular spring being taken 
as unit}'. 

In all the formulae of the table we have 

£ = the modulus of elasticit}', 

2 
G = the modulus of torsion = — .£, (see I 13). 

The coefficients for the resistance of the materials used in 
springs will be foimd in J 2. It must not be forgotten that for 
materials used in torsion, to obtain the same security as when 
used in flexure, the pemiissible stress 6' should be f its usual 
value (see g 5). The formulas are intended only to be used 
when the force P is applied as shown iu the figures. 

The volume F of an)- form of spring is according to the 
formula : 



V=C.{P./]^ 



(40) 



in which C is a constant depending upon the form of the spring ; 
while P/^ is the product of the load into the deflection, or the 
so-called work of the spring. This shows the interesting fact 
that springs of the same general form and same material are 
alwaj-s of the same weight for the same work, without regard to 
the actual length or proportion of dimensions. 




Rectangular 
Spring, 



Simple 

Triangular 

Spring. 



Compound 

Triangular 

Spring. 



Supporting Power. 



5- M2 

~6~ T 



S Ik'- 



S iUfi 
-. No. of plates. 



Deflection. 



/=6- 



f=('' 



/=6- 



PP 



Elasticity. 






j_ 



S I 

E h 



I 



Remarks. 



An approxi- 
mation to 



^^Z- 



will be secured 
by making' the 

2 
end = h. 



Body of uni- 
form resistance 
to bending. 

In practice the 
end is made 
somewhat thick- 



This is equiv- 
alent to a sim- 
ple triangular 
spring, with a 
base = i b, as 
shown by the 
dotted lines. 



THE CONSTRUCTOR. 



19 




Flat Spiral 
Spring. 



Flat Helical 
^ Spring. 



Round Helical 
Spring. 



Simple Round 

Torsion 

Spring. 



Simple Flat 
Torsion 
Spring. 



Helical Spring 
of Round Wire 



Helical Spring 
of Flat Wire. 



Conical Spring 
of Round Wire 



Flat Volute 
Spring. 



Supporting Power. 



bifi 



'32 ~R' 



P = - 



i'-A'- 



3 J! ^m + m 

Approximately when 
5 V- A2 



R 3 {0.4^ -}- o,g6/i) 



P=- 



b"-Jfi 



3 '^ s/f- + h'- 
Approximately when 

5 b°- >:'- 

P = 



R 3 (o.4(5 + 0.96/1) 



_ ^S f J^L_ 

3 R ^Ifl. + hr- 

Approximately when 
h>b. 

s m m 



p = 



R 3 (0.4^ + 0.96A) 



Deflection. 



/=R^ = ii : 



/=i?^=I2- 









/=R9- 






f-^ 



32 PF"-l 



Gdi 



f= 



Approximately 
16 PR'-l 



/=- 



Gdi 



Elasticity. 



R 



S 
E 



JL 
R 



_S_ 
E 



R 



S I 
E ~d 



R 



S I 
G d 



R G 

Jh 



f 



s I 

G d 



R ^ G ' 

ly/ b"- ■\- ffi 



R 



S I 
G d 



Approximately 

__3_ PR"~l h"- + h'2- 

~ ~2 'G~ ' i^ B 



R 



Kemarks. 



/ = the devel- 
oped length of 
the spiral. 

All three forms 
of uniform re- 
sistance. 

The va.ue -r- 
R 

is the a- gle of 
rotation -^ , pro- 
duced by die 
load P. 



Cases VII. to 
X. are bodies of 
uniform resist- 
ance to torsion. 



Springs of the 
form of VII. and 
VIII. may also 
be combined 
into compound 
forms. 



In cases IX. 
to XII. / is al- 
ways the devel- 
oped length of 
the spring. 



It is immate- 
rial whether the 
breadth of the 
plate is parallel, 
normal, or ob- 
lique to the axis. 



Here, as in 
case XII., also 
the spring is 
measured to the 
apex of the cone. 

The weakest 
point is at B. 



By making a 
gradual reduc- 
tion in the value 
of /:, from B to 
the end, this 
may be made a 
form of uniform 
resistance. 



20 



THE CONSTRUCTOR. 



The quotient —^, shows that a small modulus of elasticity, 

■when combined with a high modulus of resistance, indicates 
the best material for the construction of springs. According to 
the table in \ 2, we have : 



Hardened and tempered steel -t^ 
Ordinary steel (not hardened) 
Brass 
Wood 



42,600,000 
(go, 000)^ 
28,400,000 

(35.500)' 
9,230,000 

(6,816.'^ 
1,562,000 

(2,840)^ 



.00052 
.00223 

=: .01986 
= .01936 



This shows that hardened and tempered steel is theoretically 
the best material of springs. It is also worthy of note that in 
all the examples given, the deflection is proportioned to the 
load. It follows from this fact that the time of vibration which 
any of these loaded springs possesses, is of the so-called "sim- 
ple" character, of the same nature as that of a pendulum. 
Neglecting the weight of the spring itself, we have for the vibra- 
tion of a loaded spring the same rate as that of a simple 
mathematical pendulum of a length equal to the deflection of 
the spring/, which is 



in which ^ is the acceleration of gravity = 32.2 ft. 



(41) 



Examples on the theory of springs : i. Given a simple triangular spring, as in case 
II., for a load P = no Ib's., and a deflectiony"= 0.7S". Taking the material as cast 
steel, with E ^ 42,600,000, and making S, the greatest permissible stress, =^ 56,800 
lbs., and also taking the length I ^= 15.75", we then have 

t E ' h 



from which 



Substituting this 



we get ; 



The 



0.78 
"5-75 



56,! 



j^ ^ 56,800 X 15-75 X 15-75 
0.78 X 42,600,000 
I the formula" 



Eilfi 



or & = t 



!,6oo,c 



E/lfi 



15-75 



h 

olume 7 



6 X 110 X (15-75)" 
42,600,000 X 0-78 X Co-424)' 

bill 1. 018 X Q-424 X 15-75 



= i.oiS" 



Example 2 : If we keep the same conditions, but make the length 11. 8", we shall 
have 



/i = 



56,800 X 11-8 X ii-B 
0.78 X 42,600,000 
6 X no X (ii.8)» 
42,600,000 X 0-78 X (0.238)3 



■ 0.238" 



3-39 cu. in., thuscon- 



rp, , . ^u . ,r l^Ill 2.42 X 0.238 X II S 

The volume m this case ^= V ^ = ^ ^ ^ '^ ^ 

2 2 

firniing the remarks on formula (40) by showing that the volume depends upon the 
load and the deflection, and is independent of the proportional dimensions 

E.\ample 3 : Let us now suppose the same conditions to be applied ti a helical 
spring such as No. IX , also made of cast steel Since this is a torsion spring, in 
order to obtain the same security we must make 5— -i- of its preceding value, or 
— — - 56,800 = 45,440; and the wire may be taken as 0-24 in diameter. 

We then have from the fable 



P-^ — —, or 110 = 



from which we get 



45,440 
16 



(0.24)3 



R. 



^ 45,440 X 3 1416 X (0-24)3 

„, , 16 X no ^ "'' 

The length / is obtained from column (6) 

f SI /Gd 



= 31-3" 



in which G ^ ~ E - 
5 

/^ °-7^ X 17,040,000 X 0.24 
2-X 1-121 X 45,440 
This would make the number of coils 

31.3 ^ 

2XTX1.121 '■■'" 
or about 4K coils. If more colls are preferred, the diameter, d, of the wire mu^t he 
reduced and the calculation repeated. ' ' °' 



■ 17,040,000. 



^ ^itR~ 



The volume l\ 
This gives the ratio 



4 
l\ _ 1-416 

"7 ^ 0-410 or 

3-4 



V 



31-3 X 0.7854 X (0.24)2 = 1. 416 cu. in. 

as given in the table. 



Example 4: Torsion springs have recently been applied to railway c 
1k™±°I" '":.l?'5:;':h'=iLl'.'^? design of an American, Mr. DnX; 



/ cars in the 

shaped spring is belit'at the ends into two°eibmvsr.^'srwhi'ch "are ^Urhed bl'^K^lV 

to a block which rests on the axle bo.x. A saddle A transmits the n^H„f, I 

to the spring, while the other end is supported at C by a hoTk °^ *" "^"^ 



In prder to determine the stress S, in one branch A C, of such a spring, let us takfr 
the diameter of = 1.14", and the lever arm, R, which is the horizontal projection of 
A B, as 4". The load on the spring is one-fourth the load on the car, 22,000 lbs. + 
one-fourth the weight of the car itself, 18,000, and one-half of this is borne by each- 
branch of the spring, making the load at the end of the lever R in this case to be- 
;,3oo lbs. 
In the preceding table, under case VII., column 4, 
16 PR 16 5000 X 4 
~ "'',750 



IT d^ IT (1.14)^ 

If this spring is made of Sheffield steel which has a modulus 



)f elasticity E =^ 



24,140,000, then the modulus of torsion G '= ~ E ^ 9,656,000. 
If the length /= 33.5", the deflection, according to column 6 in the table, will be 
/= ^-^■^'^ ^ 2 X4X 68.750 X 33-5 _ J 6 ,/ 
Gd 9,656,000 X 1-^4 

The above described spring weighs 24.2 pounds, which is about -^l-^ of its gross- 
load, or about n^r of its net load. 




Fig. 9, 

A double armed plate spring of the form No. III., to have the same supporting. 

pc vver would weigh about a hundred pounds, or jj ^ its gross load and '-^ its net load. 

As long ago as 1857 1 called attention to the superior economy of torsion springs- 
over plate springs for railway use. The principal reason for the tardiness of railway- 
men in appreciating this fact may have been partly due to the difficulty of securing- 
a proper temper in the round steel, although this seems to have been entirely over- 
come in the case of the Dudley Spring. 

In the little pamphlet on "'The Construction and Calculations of Springs," which 
I published at that date, the comparative weight of the torsion springs VII. and IX., 
and the triangular plate springs 11. and III., is given as /j^, instead of -{v: as in the^ 
preceding table, but the latter is snown to be more nearly correct in practice. 




In Figure 10 is shown the manner in which a helical spring may be applied to the 
bearing of a goods wagon in the place of a plate spring. The box is guided in the 
frame B, B, and the spring D is interposed between the sill A of the wagon and the 
journal box C The form of the spring is a single helix with the lower end flattened 
for about y^, of a turn in order to give a fair bearing in the cap E of the box. The 
upper end sc^e^\'s for about ij^ turnsinto the cap /'.where it is clamped by thescrew 
G after the load has been equalized, and in this way any desired adjustment may he- 
secured. 

An example will probably be the best method of showing the manner of calculat- 
ing such a spring. 

An ordinary German four wheeled goods wagon weighs about 11,000 pounds, and 
carries about 22.000 pounds load. This gives about 8,250 pounds to be supported by 
each spring. We will assume a deflection of 1'%", with a permissible fibre stress S 
of 6S,ooo lbs., and take G = 9,656,000, as before. 

Since it is desirable to use such diameters of spring steel as correspond to commer- 
cial sizes, it is better to select a diameter ^for the steel, and deduce a corresponding 
radius R for the helix, according to the formula for case IX., col. 4, page 64, 

IT d^ 

10 P 



777^ CONSTRUCTOR. 



21 



We will take the successive cases in which the diameter d of the steel is i", ij'b"* 
tYs" and i^V'- 

Now for these respective values we must select such a number of coils, «, that 
■with a load P= 8,250 lbs., we shall get a compression^"^ i-75"- 

We have for n the fact that 2ttRh = the length / of the uncoiled spring. Substi- 
ituting this in the formula fory", case IX., we get 

^~ ~ ~G 'd" 
som which 

/G d 

Now the least possible distance between the cap F and the socket E is nd -}- r, 
-and we must also provide for a space u between the coils, say 0.3". This gives the 
-distance ^ from centre to centre of coils of the unloadea spring 

nd +y + Iter 
n 

The tot^.l height of the spring, however, will be greater than ns by 1.5^ + d, since 
3J4 coils enter into the cap F, and one-half the diamater of the steel in the last coil 
Tnust be added both top and bottom. Adding to this the thickness 5 of the cap and 
the socket, as J4". we obtain the entire height occupied by the spring and its fittings. 
■Of course this height is limited by the space at our disposal between A and C, and 
in this case it may be taken at 14 inches, and in any case it will generally be depen- 
-dent upon other circumstances. Substituting these values in the several formulae, as 
^iven above, and tabulating the results, we have: 



d = 


i" 


ifs" 


iKs" 


iiV 


R = 


1. 610 


1.94 


2.30 


2.71 


or say = 


m 


2.00 


21% 


'H 


A = 


7-5 


5.6 


4.18 


3- 19 


K<f — 


7-5 


5-?§ 


4.70 


3'? 


fla = 


2.25 


1.6S 


1. 25 


0.90 


/ = 


1-75 


1-75 


1-75 


1-75 


» (rf + cr) +/ = 


11.50 


9.38 


7.70 


6.49 


J. = 


'■A3 


167 


1.84 


2.03 


or say = 


I'A 


1^8 


lif 


2.00 


1.5-r = 


'H 


2,^ 


2X 


3.00 


26 == 


1. 00 


1.00 


1. 00 


1. 00 


Total height = 


15-75 


13.82 


12.45 


11.49 


or about = 


I55i 


13% 


I2j4 


11}^ 



It will be seen that the first size is too high for the space at our disposal, but that 
■the others may be used as circumstances may dictate, all lour springs having the 
:same compression and stress upon the material. The simplicity of this construction 
is noteworthy, and the economy of material is noticeable. For passenger coaches, 
where more elastic springs are desirable, _/" may be made from 2 to 3 inches, and a 
<:ombination of several springs may be used. 

The use of vulcanized rubber for springs and buffers is now 
quite general, usually in the shape of rings or collars between 
plates of iron. The resistance of rubber to distortion has not 
yet been fully investigated in an experimental manner, but the 
following examination of buffer springs may be of service ; the 
■data being from the valuable researches of Chief Ungineer 
"Werder, of the establishment of Klett & Co., of Nuremberg. 

The usual sections of buffer rings are shown in Figs. 1 1 and 
12, in which one of the plates carries an annular projection, and 
the other a corresponding depression, between which the springs 
■are held, and lateral motion in the buffer case prevented. 

When the ring of rubber is under compression its volume is 
Tinchanged ; the cross section is reduced, but the diameter of the 
ring is increased proportionall}'. The elements which are sub- 
jected to the greatest strain lie at the circumference £, and are 
under tension, as is proved by the cracks which appear when 
the limit of elasticity is exceeded. 

The limit of elasticity will be reached by a load of about 700 
pounds to the square inch of original cross section normal to 
the axis ; or, in other words, a modulus of compression T= 700 
may be taken. This modulus is slightly higher for rubber of 
the lightest speciSc gravity (about 7S0 lbs. ) , and less for a heavier 
specific gravity (about 640 lbs. ) . The specific gravity 7, which 
depends upon the proportion of sulphur, varies from i for the 
lightest, to 1.15, or even 1.32, for the heaviest. 



When the elastic limit is reached, the middle section E F of 
the ring is double its original area, while the periphery is 
^ the original periphery A B C D. 

The compressibility within the elastic limit is dependent upon 
the quality of the rubber, and may be approximately determined 
by the following empirical formula : 



/I = 0.026 



w 



(42) 



in which : A = the extent to which the spring is compressed by 
a load P\ I = the original thickness of the spring ; g = the 
original cross section in a plane normal to the axis ; and y = the 
specific gravity of the material. 




Fig. II. 



Fig. 12. 



Example : A buffer spring similar in shape to Fig. ii has an outside diameter of 
^yz", and an inside diameter of 2^§", which gives an area of cross section q of 18.07 
square inches. The thickness of the uncompressed ring /^if'^", and its specific 
gravity y = i. The load to be supported is 5500 pounds, which corresponds to a 

P 

pressure per square inch = 305 pounds, which is well within the elastic limit. 

? 
According to (42) the compression will be : 



A =i 0.026 X 1-375 V 



305 — 0.624, say Ys" 



The Belgian engineer, Stevart, has also made extensive re- 
searches upon the subject of the resistance of rubber. These 
experiments appear to confirm the opinion that any change of 
shape is unaccompanied by any change of volume, and that 
rubber is practicallj' as incompressible as water. The experi- 
ments on tension gave a modulus of elasticity of 1 19. 28 lbs. In 
regard to compression Stevart deduced a formula similar to the 
preceding : 

7^1= ^aP+i; 

in which a is a coefEcient dependent upon the form of the 
spring, and determined experimentall)-. In the case of locomo- 
tive buffers, which are composed of several rings, the compres- 
sion of each ring should be computed separately', and their sum 
taken. 

Rubber springs are used very extensively, but the principal 
objection to them is that the material gradually loses its elas- 
ticity and becomes a hard, unyielding mass. It has been found 
that this is largely due to friction between the rings and their 
cases, aud great care should be taken that boxes for rubber 
springs should have ample allowance made for the increased 
diameter of the spring when compressed. 

It is a matter of importance to choose such a shape for a 
rubber spring that it shall not have a tendency to form puckers 
in the edge when it is under pressure. This is shown iu Fig. 
12, where the slight concavity in the edge would soon develop 
a crack when compressed, as at F^ F^, while the shape in Fig. ii 
has no such tendency. 



22 



THE CONSTRUCTOR. 



SECTION II. 



THE ELEAIENTS OF GRAPHOSTATICS. 



?2I. 

Introductory. 

The equilibrium of forces may be very clearly shown by the 
graphical method, since it is possible to show the direction, ex- 
tent and position of any force by a right line. The direction 
of a force is determined by the angle which its representative 
line makes with the horizontal axis of co-ordinates ; the length 
of the line gives the absolute amount of force exerted, the alge- 
braic character of the force (plus or minus) being indicated by 
arrow-heads ; while the position of the line in the system of co- 
ordinates, makes it possible to show any constants which 
may occur in the equation of any right line. This repre- 
sentation of forces by means of geometric magnitudes makes it 
possible to solve problems in statics entirely by means of 
geometrical constructions, and in many cases this will be found 
simpler and more convenient than the use of algebraic analysis, 
especially in those cases in which the values to be determined 
are themselves geometrical quantities, and therefore are to be 
drawn when found. The various details of the method have 
been arranged and collected into a system which has been called 
Graphical Statics, or, as we have here termed it, Graphostatics.* 
This method is of especial value in the study of Machine De- 
sign, and in the following sections of this work many applica- 
tions of it will be found. It is for this reason that the following 
brief exposition of the leading principles of the method have 
been here grouped together. 

There is a distinction to be made between Graphostatics, 
properly so-called, and the mere graphical calculations of simple 
values, considered merely as magnitudes. This is more properly 
to be considered as graphical arithmetic, or Arithmography.f 

In the following pages this branch of the subject 'is not very 
fully discussed, only an outline of its application to pure arith- 
metic being given. It will be found, however, to be a subject 
of much use to the mechanic, as many examples of its applica- 
tion in future pages will show. 

? 22. 

MnLTipi,icA.TioN BY Lines. 

In graphical calculations dimensions are taken with the 
dividers and scale, and any convenient unit may be selected, 
such as the inch, millimetre, decimeter, square foot, cubic foot, 
unit of velocity, unit of money, etc , etc. It is readily apparent 
that the operations of addition and subtraction may be per- 
formed by simply marking off the various values upon any line. 
The operation of multiplication is not quite so simple, and a 
brief explanation may be necessary. 

In all cases it is of course necessary that the same unit must 
be chosen for all the quantities involved, and this holds good for 
multiplication as well, and the same unit must be used to meas- 
ure the result as has been chosen to express the original quanti- 
ties. If, now, we wish to multiply two lines, a and b, together, 
or, more correctly, to multiply a line of the length a by a line of 
the length 5, we must find a line ,v, which will contain our 
chosen unit, a y, b times. This is a simple operation, and may 
be performed in several ways by means of similar triangles. 

I. Draw O E, Fig. 13, horizontal, making its length equal to 
unity ; erect sXE & perpendicular, and intersect this from O with 
O B=ib. Lay off O -4 = a, and from A draw a parallel to E B, 
intersecting O B produced at C. Then O C will be the desired 

O C O fi 
product X. That is to say, -^--^ = — ^, and since C £■ == i, we 

* See Culmann, "Graphical Statics," Ziirich, iS66. 

t See " Principles of Graphical .Arithmetic," by Dr. Eggers, Schaffhausen, 1865 ; 
also Schlesinger, " Power Curves," in Journal of the Austrian Society of Engineers 
and Architects, 1866; also E. Stamm, "Graphical Calculus," Proc. Royal Inst., 
Lon.bardv, Vol. VI. 



have X = — . This solution requires that one factor, b, shall 

be greater than unity. 

II. Fig. 14. A modification of the preceding may be made 
by drawing E B inclined, instead of perpendicular to O E, in 
which case both factors may be less than unity. 

III. We may make, as iu Fig. 15, O E and O B a.s before, 
produce O A^=a,anA drawee, so that the angle OAC= 
OEB,ao that A C will be the anti-parallel to E B. Then O C 




Fig. 13. 



Fig. 14. 



will be the desired product x, since the triangles O E B and 
C^ Care similar. This anti-parallelism is shown by the fact 
that OE' = OE, OB' = OB, a-aA A C is parallel to E' B' . 
If the triangle B E' B' is rotated to the right about an axis, 
passing through B B' , the two triangles, B B' E' and B B' E, 
will form a parallelogram ; hence the term anti-parallel. This 
construction is most convenient when E B is perpendicular tO' 
O E, which can only occur when b is greater than i. 

IV. We may make, as in Fig. 16, O E= unity, lay out on 
OEfhe factor O A ^ a, and erect a perpendicular or inclined 
line at E, in which E B= b ; then draw through A a parallel 
to E B, and this latter line will intersect OB, prolonged so that 




Fig. 16. 



Fig. 17. 



Fig. 18. 



A (7== X, since CA ■.OA^=BE:OE, or a- : a = 5 : i ; a and b 
being either greater or less than i. Now make E B^^ 5,, and 
draw O B^ to intersect CA, prolonged at C^, then A Ci = x^, 
the product of a and b^, and C C^ will be the product of a into 
B B^, or : 

X -[- x-^==a[b-\- bi). 

Of course, the factor b, which is to be multiplied by a, may- 
extend on both sides of the base line O E, and the desired pro- 
duct, ab = X, will then be the distance on the parallel to b, 
which is included between the two lines drawn from O through, 
the extremities of b. 

V. In Fig. ij O E ^ unity, i? i? = the factor b, and O B any- 
yalue, so that OB<^OE -[- E B. Lay out O A on O B. making 
it = the factor a, and draw from A an anti-parallel to E B (see 
III.), then A C will = x. Vor A C : O A = B E : O E, or 
X -.a^b : J, and a and b may both be less than i. 

VI. Again, we may make Fi.g. 18 O E =: i, erect a perpen- 
dicular at E, make E A ^ a, E B = b, join O with B, draw 
B B^ normal to OB, and draw from A a parallel to B B\ then 
will EC^ the desired product x. For v/e haye E C : E A = 
BE: O E OT X ■.a=: b : 1. 

It often occurs in designing that we have already a diagram 
drawn which may serve for a portion of the construction, and 
in such cases the following methods may be found convenient. 

VII. Fig. 19. O A = a and B' B ^ b are either at right 
angles or inclined to each other, so that B' falls between O and 
A. Layout on OA the unit Oil, join i? with E, and draw 
from A a parallel to B E, and from the point C, where it inter- 



THE CONSTRUCTOR. 



sects O C draw C C parallel to B B' , then CO will = x for 
we have C O : O A — B B' : O E ox x ■.a = b w. 

VIII. Fig. 20. Given as before, O A = a and B B' = b, either 
perpendicular or inclined to C.<^. Draw O j5' parallel to B B', 
and equal in length to unity ; join £ to A, and draw from B a 




Fig. 19. 



Fig. 20. 



Fig. 21. 



parallel to E A. This will cut off on O A prolonged the dis- 
tance 5 ' C"= .*•, ior B' C : B' B = O A : O E or .X : b = a : 1. 

IX. Fig. 21. Given A A' = a and B B' ^= b perpendicular. 
Draw A B, and prolong it until it intersects at jE a line drawn 
parallel to A A' at a distance O E ^ i. Join E A', and draw 
from B a parallel to it, cutting A A' at C, then will A C =: ,v, 
for A C : C B = A A' : A' E, andi A C : B' B = A A' : E O, 
ox X : b = a : 1. 

X. Fig. 22. Given A A' = a and B O = b, perpendicular 
to A A'. Open the dividers to O E =■ i, and intersect A A' at 
E. Draw from A' a. parallel to O E, and from A a normal, the 





Fig. 22. 



Fig. 23. 



two lines intersecting at C, then A C= the desired product x. 
For, since the angle C A A' ^ B O E, we have A C : A A' = 
O B : O E, ox X : a = b : 1. The line A A' is in this case pro- 
jected upon a perpendicular to O E, or is what is called the 
anti-projedion of A A' to O E* 

XI. Fig. 23. When a and 5 intersect each other at right 
angles, as in the figure where A A' = a and B O =^ b, then 





Fig. 24. 



Fig. 25. 



draw from B a parallel to A A', and mark off with the dividers 
from O, O E ^ I. Draw A' C parallel to O E, and A C normal 
to A' C then A C= .r, for since the angles at E and A' are 
equal, we have A C : A A' = O B : O E, ox x : a = b : 1. 

The continuous multiplication of several factors may be 
accomplished by combining the preceding methods in various 
■ways. 

Suppose we desire to obtain the product of three lines, a,b, c, 
■we may iirst find, according to I, the product .r, ^ a b (Fig. 24), 
transfer O C = a b down to O C upon O A, draw from O the 
line O £> = ir, erect from C a perpendicular, and prolong O D 
to F, and O /^will be the desired product, x = a b c. 

Or we may make, as in Fig. 25, after having found O C = a b, 
draw E D = c (Case IV.), and prolong O D until it intersects at 
jpa perpendicular from C , when C F= x, 

i 23. 

Di'visioN BY Lines. 

Division may readily be accomplished by reversing the 
methods employed for multiplication. To divide a line a by a 
line b, we must find a third line x, which must contain the unit 



of a and b, -^ times. From the previous examples we may de- 
rive the following methods of division. 

I. Fig. 26. Make O E = unity, erect at E a perpendicular 
or inclined line, intersect it with the divisor O B = b, prolong 








A 




c/ 


X 


1 


I 


^ 


:! .h 



Fig. 26. 



Fig. 27. 



Fig. 28. 



O B, and make O A = the dividend a. Draw from A a paral- 
lel to B E, and its intersection with O E prolonged will give 
the quotient x. For we have O C : O E = O A : O B, that is, 

, a 

X : 1 = a : b, or ;ir = -r-' 

II. Fig. 27. Make O E = unity, also lay off on O £■ the dis- 
tance O B ^ the divisor 5, erect at i? a perpendicular, and in- 
tersect it from O with O A ^ the dividend a. A perpendicular, 
erected from E, will then intersect O A at C, and (9 C^ .i', for 
we have again O C : O E = O A : O B ox x : i ^ a : b. 

III. Fig. 28. Make O ^= the divisor b ; on O B lay oS O E 
= I ; at j9 erect a perpendicular A B = the dividend a ; join 
O A. Erect at ii a perpendicular, and it will intersect O A at 
C. Then E C = x,{ox E C : O E = A B : O B, ox x : i = 
a : b. 

§24. 

Multiplication and Division Combined. 

When it is desired to multiply a number a into a fraction — , 

the operation really consists in mnltipljdng a by b, and dividing 
the product a x b hy c, in order to obtain the result x. If we 

recollect that for x = , we may write x : a = b : s, we will 

see at once how the combined operation maj' be performed by 
making the distance O E equal to the denominator r, instead 
of unity, as heretofore. We will then be multiplying the line a 

by the ratio -^.instead of . The following illustrations will 

make the operation clear. , 

I. In order to multiply a quantity a by a fraction — , we make, 

in Fig. 29, O A ^= a, lay off on O A, O E ^ c, erect at ii a 
perpendicular, and intersect it at B, with a distance from O 




c 



Fig. 29. 



Fig. 30. 



* See Culmann's " Graphical Statics.' 



equal to b\ then prolong O B until it intersects at Ca line 
drawn from A, parallel to E B. Then O C will equal x, for 

we have O C : O B ^ O A : O E, ox x : b ^ a : c, ox x ^ — . 

a b '' 

II. If we wish to find the product , we make, Fig. 30, OA 

== a, and make the distance O E ^ twice the unit of measure- 
ment, draw E B = b perpendicular to O E ; draw a line from A 
parallel to E B, and prolong O B until it intersects this last 
line at C. Then A C will be the desired product x, for 

A C : O A = B E : O E, ox X : a = b : 2, ox X = '!^. 

' 2 

These methods, which may be extended much in the same 
manner as the various methods of multiplication given in 'i 22, 
will be found of great service in the graphical calculations of 
areas, as we shall see. 

?25- 

Area op Triangles. 

Since the area of a triangle is equal to the half-product of its 
base and altitude, it is readily calculated by the method given 
in the preceding section. 

I. Fig. 31. Selecting the side O B ^ b cf the given triangle 
O ^ i? as a base, which gives the perpendicular A A' = tU2 



24 



THE CONSTRUCTOR. 



height //, although this line need not be drawn, we mark off the 
distance O E ^ 2, units (inches, decimeters, etc.), and draw 



III. Fig. 37. The diagonal A C=b divides the figure ABCO 
into two triangles, the sum of whose heignts = O O', which is 





Fig. 31. 



Fig. s2. 



from B a line B C, parallel to an imaginary line A E. This 
line B (Twill intersect the side O A prolonged at C, and a per- 
pendicular dropped from (T to OB, will give C C ^ = the 

desired area/ (see VII., ? 22, and II., \. 24). 

II.- Fig. 32. From the end of the base line OB draw the 
perpendicular O E ^2 units, draw the altitude A A' ; also draw 
from A the line A C parallel to E B. This will cut off on the 

tase line the distance A' C, which is the product / = — . 
(? 22, VIII., and I 24, II.) _ _ ^ 

III. Fig. 33. Prolong the base line iJC and the side BA 
until the vertical distance between them O E ^ 2 units. Join 





Fig. 33. 

E to C, and draw from A a. line parallel to E C, intersecting 
the base at B,andB£> = — =/. (? 22, IX., and § 24, II.) 

IV. Fig. 34. From the vertex O, with the dividers open a 
distance equal to 2 units, intersect the base at E, and make the 
anti-projection of the base A B by drawing B Cparallel to O E, 
and A C normal to B C. Then A C= the product of the base 
b, and one-half the altitude O O' ^ It, and hence is the desired 
area/of the triangle. (§ 22, X., and § 24, II.) 

If the unit is taken as one inch, the value of the area y"%vill 
be given in square inches, or if a decimeter is taken as the unit, 
the area will be in square decimeters, etc. 

If we findy= y, the area of the triangle is seven-eighths of 
a square inch ; or if it measures 72 millimeters, the area would 
be 0.72 square decimeters, or 0.70 x 10,000 = 7200 sq. mm. 

§26. 

Area of Quadrhaterai, Figures. 

In determining the area of a quadrilateral figure, it is either 
obtained directly, as in the case of a parallelogram ; or it may 








Fig. 35. 



Fig. 36. 



be separated into triangles, which are measured separately ; or 
the figure may be reduced to its equivalent triangle. 

I. Required the area of the parallelogram ABCO, Fig. 35. 
Taking the side O A &s a. base line, lay oS O E = unity, and 
erect the perpendicular E E' = h. Prolong O E until it inter- 
sects a perpendicular from A at D, and the distance A D will 
be the area of /= A A. (g 22, IV.) 

II. The quadrilateral figure ABCO, Fig. 36, may readily 
be replaced by a triangle of equal area by drawing the line O A' 
parallel to the diagonal O B, for since the triangle O A' B is 
equal in area to O B C, we have the area of the triangle O A' A 
is equal to the area of the figure ABCO. Now, according to 

IV., I 25, we make O E = 2, and draw A D, the anti-projectioa 

of A A' and A D = f, the desired area, 





Fig. 38. 

the anti-projection which O B makes on A C. The multiplica- 
tion of O O' \>y — may be made according to XI., \ 22, and 

II., ? 24. Draw O' B E parallel to A C, making O E = 1, also 

draw A D parallel to E O, and C D normal to A D, then C D 

=/= the area of A B C O. 

IV. Fig. 38. The figure ABCO may be converted into a 

triangle whose altitude = 2, when the base will be equal to the 

h b 
product — . From O describe an arc with a radius O E ^= 2, 

and draw a tangent passing through an angle of the figure at B, 
opposite the angle O. From the other two angles, A and C, 
draw lines parallel to the diagonal O B, intersecting the tangent 
at A' and C A' C will then be the base of a triangle whose 
altitude = 2, and whose area is the same as the figure ABCO, 
and the area_/'= A' C . Many similar methods may be deduced 
from the preceding examples. 

?27. 

Area of Polygons. 

The area of a polygon is measured by reducing it to its 
equivalent triangle. This may be done in the following manner : 
From the angle O of the polygon O A B C D E, Fig. 39, draw 
a diagonal O B to the next angle but one, and then from the 





Fig. 39. 



intermediate angle A draw A B' parallel to O B, prolonging 
the third side B C to B' . If we join G B' , we have the triangle 
O B B' = O B A, and hence the figure O B' C D E will have 
the same area as the original figure, but will have one less side. 
Then join O C, and draw B' C parallel to O C, and so we may 
proceed until we have obtained a triangle O C D' of equivalent 
area to the original figure, and whose area maj' be determined 
by any of the preceding methods. 

Regular polygons, such as the hexagon. Fig. 40, only require 
half the operation to be performed, and then the area measured 
as a parallelogram. 

?2S. 

Graphical Calculation of Powers. 

A line a, raised to the «"' power, reall}' means the determina- 
tion of a line x whose length shall contain the unit of measure- 
ment rt« times. The following methods are applicable when a 
is a positive or negative whole number, and the process is really 
a repeated application of the multiplication of a by a. As in 
the previous cases, this operation may be performed in various 
ways. 

I. (See \ 22, I.) In Fig. 41 make O E ^ unity, erect at E 
a perpendicular, and intersect it at A^ with the distance OA^^a, 
the original factor. Carrying this distance O A-^ down to B^, 
and erecting a perpendicular at Bi, we get O A^ = a' (see I., 
? 22). This again carried down to B,, and a perpendicular 
erected at B,, gives O A^^ a^, and so O A^^ a*, O A^ = a^, etc. 

If we lay off O Bm, equal to any power of a, say a"', and erect 
perpendicular at Bm, the intersection with O A^ prolonged will 
give the value of «'" -F '. Again, if we drop a perpendicular 
from the end point Am + i of any power of a to the axis O E, 
it will cut off a distance Bm, which will be the next lesser 
power of a (see I., I 23). 



THE CONSTRUCTOR. 



25 



The perpendicular A-^ E, from A^ upon O E, gives the first 
jower a'. If we now make O Aa = O E, and drop the perpen- 



Ani« 




Fig. 41. 
•diculary^o j? — i, we have OB — i = a — ■, which = — , which is 
the reciprocal of O A-^ ; in the same manner we get O B —z^^ 



a' ' a' 

II. By combining the methods of multiplication I. and III. 
■of § 22, the following method for powers is derived. In Fig. 42, 
jnake O E = \, O Ay ^ a, E Ay perpendicular to O E, and draw 




irom A^ a perpendicular to O Ay, cutting O E at A, ; then O A^ 
= a'. From A.^ ^ perpendicular to the base will give A, and 
O A^ = a^ \ another perpendicular to O A^ gives A^ and O A^^ 
a*, and this may be continued indefinitely for positive powers of 
a. By working backward from E, we get OA — -^ as the recipro- 
cal of a, O A—-^ = — J-, and so on for negative powers of a. 

Both the preceding methods assume that a is greater than i ; 
"the following may be used when a is less than i : 

III. In ]?ig. 43 make O E ^ i, and draw O A = a at such an 
angle that A E is perpendicular to O A. Erect the perpendicu- 

K 





Fig. 43- 

lar E I, and continue with the alternate perpendiculars i 2, 2 3, 
3 4, etc., and we have : O i ^ — , 2^ — =-. O x = — r-, etc. 

Working to the left from E in a similar manner, we get 
O — 2 ^ a^, O — 3 ^ a^, O — 4 ^ «*, etc., the positive powers 
"being to the left, and the negative powers to the right. 

The zigzag lines which are thus drawn back and forth between 
the two axes have a relation to the powers of a which may be 
utilized in the following manner : 

IV. Make, in Fig. i\d„OE=j, OA=a, and the angle 
O A E^<fi'' \ also OB at right angles to O E, and prolong E A 
to B. Now draw the alternate perpendiculars as before, and we 
liave the following values : O A = a, A 2 ^ a'-, 2 3 ^ a', etc., 

also O E= a° , E — i^a — i^ — , — i — 2 = -v. etc. 

a a' 

V. Fig. 45. Make O E ^ 1, and describe upon it as a diam- 
•eter a semicircle, make O i = a, and from i drop a perpendicu- 
lar I 2 upon O E, then C 2 = (Z- (see Problem III. of this section). 
"With C 2 as a radius from O, describe an arc, and from its inter- 
section with the circumference drop the perpendicular 2 4, and 
(? 4 = a*, and by continuing in the same manner, we get OS ^ 
a^, O 16 = a'^, etc. The intersection 3 of the radius O i with 
•the perpendicular 2 4 is, at a distance from O, equal to a?. For 



we have : O 3 : O i = O 4 : O 2; or, O 3 : a ^ a* : a', that is, 
Oz = a\ 

In this way we may prove that each line drawn from O to the 
upper extremity of the successive perpendiculars on O E, inter- 
sects the following perpendicular at a distance from O equal to 
the next less power of a. This provides a method of obtaining 
the intermediate powers of a by merely drawing radii and per- 




FiG. 45- 

pendiculars. Each newly-found power gives a radius for a suc- 
ceeding one, and the operation may be continued indefinitely, 
as shown in the diagram. 

VI. The following method is suitable for any given value of 
a, whether greater or less than i. In Fig. 46 make O E ^ 1 
on the axis X O X, erect a perpendicular at O, V O V, and 
mark off O A ^ a. Join A E, and draw A 2 normal to E A, 
and it will cut off on the axis of X, a distance O 2 = a' ; then 
draw 2 3 at right angles to A 2, and we get on the axis of Y, 




Fig. 46. 

C 3 ^ rt', and on the axis of A', (9 4 ^ f?*, thus getting the even, 
positive powers of a on the axis of X, and the odd powers on 
the axis of }'. By carrying the spiral in the other direction we 
get the negative powers in a similar manner. Joining A E, we 

have O E = a" ^ 1 ; from that we get O — i = — , and in a 

II " 

similar manner ^, -^, etc. (See 3 22, VI.) This method is 

very suitable for showing a succession of powers in a single dia- 
gram. 

§ 29. 

Powers of the Trigonometrical Functions. 

The methods already . given for the determination of the 
powers of numbers are also applicable to the powers of the trigo- 
nometrical functions with but slight modifications. 

I. Powers of Sines and Cosines. Fig. 47. Make O E =: 1; 
the angle E O A = (p, the angle the powers of whose functions 
are to be determined, E A being at right angles to O A. Draw 
also the alternate perpendiculars ^ 2, 2 3, 3 4, etc., and E — i, 
— I — 2, etc. Then O A = cos <p, O 2 ^ cos ^ (6, O s = cos^ <p, 

O A^ cos* * ; C — I = , O — 2 ^ r-, etc. 

cos 9 cos '<p 

By drawing the alternate perpendiculars^ //., //. ///., ///. IV.; 

O — /, — /, — //, etc., we also get A E ^ sin <p, A 11^ sin^ <p, 

II, III = sin* ^, ///, IV = sin' f,0 — I= r^, —1—11 = 



sm 5* 



-, etc. 



sin'^ ^ 

II. Powers of Tangents and Cotangents. Fig. 48. Make 
E O ^ \, and O E A = li. Draw from A the spiral of perpen- 
diculars as in v., I 28, and we get the following values : O A ^ 
tan (J, I = tan' ,p, A 2, = tan' (p, etc. O E = i = tan " ?>, 



26 



THE CONSTRUCTOR. 




.-^ 



Fig. 47- Fig. 48- 

O — I ^ cot <?, O — 2 = cot2 1^, etc. This method shows very 
clearly the convergence and divergence according to the sign of 
the power under consideration. 

§30. 
Extraction of Roots. 

The extraction of the square root is readily performed by the 
graphical method, as will be seen at once when it is remembered 
that %/ a is a mean proportional between a and i . The previ- 
ously described methods for powers also suggest methods fo*- 




Fig. 49. Fig. 50. Fig. 51. 

the extraction of roots, and the three following cases will suffice : 

I. In Fig. 49 make O E ^ 1,0 A ^= a, describe a semicircle 
on O A, erect a perpendicular at E, intersecting the circumfer- 
ence at C, and join O C, then O C = x ^ •■/ a (see ^ 2S). In 
this case n > i, but in the following case a < i. 

II. Fig. 50. Make O E = s, O A ^ a, describe a semicircle 
on O E, erect a perpendicular at A, and join O C, then vfill 

o c=.ir = v/;r. 

III. Fig. 51. Make O E ^ 1, and mark off on C iT prolonged 
E A ■= a, draw on O ^ a semicircle, and erect a perpendicular 
at E, intersecting the circumference at C; then will E C^ 
X = v rt. 

The extraction of the fourth root may be performed by re- 
peating the method for square root. The graphical extraction 
of the cube root, fifth root, etc., is not so simple. Culmann uses 
for this purpose the logarithmic spiral, and Schlesinger con- 
structs a curve according to the method in § 28, but the advan- 
tages are not sufScient to warrant a further examination of the 
subject at this point. 

I 31. 

Addition and Subtraction op Forces. 

In all the preceding operations we have only considered the 
lines to represent absolute quantities, and paid little or no atten- 
tion to their direction or position in the plane of the diagram. 
The principal advantages of the graphical method are those 
which are connected with problems relating to the equilibrium 
of forces, and it is the application of the preceding methods of 
graphical arithmetic to the calculation of forces which really 
constitutes the method of graphostatics. 

When several forces are acting upon the same point, their 
resultant may be obtained by the addition of the lines repre- 
senting the forces when projected upon the co-ordinate axes. 
This addition of the projection of forces is known as graphical 
addition. This addition is performed by placing the lines repre- 
senting the forces end to end, forming a polygon, care being 
taken to avoid repeating any of the lines. If the forces, 
I) 2, 3, 4, 5, 6, Fig. 52, acting at O are in equilibrium, the sum 
of their projections will equal zero, and the polj-gon formed by 
the lines, as shown in Fig. 56, will close. The figure thus con- 
structed is called a force polygon. It is immaterial as to the 
order in which the lines are taken, as in Fig. 53 the result is the 
same whether taken in the order, i, 2, 3, 4, 5, 6, or i, 3, 4, 6, 5, 2, 
although the shape of the polygon will be different. As in 
arithmetic, the graphical subtraction of forces is the reverse of 
addition, and practically amounts to a separation of the sides of 
the force polygon into their respective forces. In graphostatics 
the forces are all taken in one plane, by projecting upon the 
plane of the diagram those forces which may be without it. The 




Fig. 52 



preceding method of addition and subtraction of lines, which 
here represent forces, but which may be taken to represent any- 
thing, is called geometrical addition and subtraction. They 
bear the same relation to geometrical multiplication and divi- 
sion as the corresponding arithmographical methods do to each 
other. These little used methods, which are of the greatest in- 
terest to geometers, we cannot discuss here. 

? 32- 
Resultant of Several Forces. 

In the preceding section we assumed that the given forces 
held eacli other in equilibrium, from which it followed that the 
diagram formed by the lines representing the forces returned to 
the starting-point and formed a closed polygon. If, however, 
the force polygon for a group of forces, such, for example, as 
the forces i to 5, Fig. 54, does not close, it follows that equi- 
librium does not exist at the point O. In order to obtain equi- 
librium it is necessary to apply a force 6 to the same point, 
whose direction and extent correspond to the line 5 6 of the 
polygon. This is the force necessary to bring the other forces 
into a state of equilibrium, and from it we also obtain a result- 
ant force /?, which is given in direction and absolute extent by 
the closing line of the polygon, but acts as an expression of the 
algebraic sum of the other forces, as shown by the arrow-head. 
From this it follows that in ever}' closed force polygon each 
single force represents the resultant of all the others in absolute 
extent and direction, except that the resultant tends to produce 
motion in an opposite direction from the corresponding force ia 





Fig. 54^ 



the polygon. In an unclosed pol}-gon the line necessary to close 
the figure gives the direction and extent of the resultant of the 
other forces, alwa3'S tending to produce motion opposed to the 
closing force. 

For example, in Fig. 54, A^ is the resultant for i and 2, and in. 
a similar manner the resultant for any of the other forces iu 
combination may be found. 

The method of representing the properties of forces by lines 
is also applicable to other quantities which possess the attributes 
of magnitude and direction, such as velocities ; also to the deter- 
mination of the path of the line which passes through the cen- 
tres of gravity of the stones of a vault, for instance ; and iu a 
figurative sense it may be applied to scientific discussions, in 
which the final result acts as a closing line to the force polygoa 
of argument. 

?33- 

Isolated Forces in One Plane- Cord Polygon. 

If lines which represent forces, and hold a bod}' in equilibrium, 
do not intersect in one point, a condition which frequently 

occurs, but have a number of intersections from « to — ( // — i ) 

2 ' 

in number, the foregoing solution can no longer be used ; but at 
the same time this more complicated case may readily be re- 
duced to the simpler form. 

For this purpose we assume the existence of a system of rigid, 
straight lines which, extending from each force to the next, 
form a polygon capable of resisting both tension and compres- 
sion iu the direction of its sides, and in which each single force 
is in equilibrium with the two forces which act along the sides 
intersecting it. A polygon formed in this manner is called a 



THE CONSTRUCTOR. 



27 



Cord Polygon, or in arch construction a thrust line, because all 
the sides are in compression, and in general such a figure may 
be called a link polygon. 

The angles of the cord polygon are called "knots." The 
link polygon m?y be used for the investigation of forces accor- 
ding to the preceding methods, when at each knot there exists 




Fig. 55- 

an equilibrium between the external forces and the stresses in 
the sides of the polygon ; for example, when the forces S^..-,_ and 
S^..^, at the knot K.,, have a resultant equal and opposed to P.,_, 
in extent and direction. The forces in the sides of the polygon 
may be called the internal forces of the link polygon. We have, 
then, for any given case two sets of forces to investigate : 
(i) the external forces, (2) the internal forces, 

since for each set there exists an equilibrium. 

I 34- 

eouii,ibrium of the externai< forces of the cord 
Polygon. 

If we take the forces /^ and P.,, find their resultant, combine 
this with ^3, find a second resultant, combine it with /*„ etc., 
we will find that in order to obtain equilibrium, the resultant 
with the next to the last force Pn — i, of the polygon, will be 
equal and opposed to the closing force Pk. This holds good so 
long as the direction and extent of the forces remains unchanged. 
From this it follows that the co-ordinate distances of the point 
of application of any force may be made equal to zero, without 
aifecting the equilibrium of the external forces. The combina- 
tion of these latter forces may then be effected in the same 
manner as if they acted at a single point. In this way the force 




Fig. 56. 



Fig. 57- 



Fig. 58. 



polygon can be used to determine the equilibrium of several 
independently-acting forces. If equilibrium exists, the polygon 
closes, and if it does not close, it shows the extent and direction 
of the force necessary to maintain equilibrium. It is practica- 
ble in this way to determine two unknown quantities in a force 
polygon. These may also refer to two forces, and may be either 
direction or extent, or, as sometimes occurs in practice, the 
direction of one force and the extent of the other. 
The following cases will serve to illustrate : 

I. Both directions given. In Fig. 56 we have the directions 
of the force lines 4 5' and A 6', and by their intersection at 5, we 
determine at once their length 4 5 and A 5. If their directions 
are interchangeable we have two solutions possible, the second 
giving the directions A VI' , and 4 V , and hence the forces 
A F"/and4 V. 

II. The extent of both forces given. Fig. 57. With the dis- 
tances equal to the extent of the two forces, we describe circular 
arcs from A and 4, and the intersection of these arcs determines 
the direction of the forces. Since the arcs intersect at two points, 
two solutions follow, giving the lines ^ 5, 4 5, and A V, 4 V. 

III. The direction of one force and the extent of the other- 
given. In Fig. 58 let the line 4 5 be the given direction of one 
force. With a radius A 5, equal to the extent of the other force, 
describe the arc shown by the dotted curve, and the two inter- 



sections give two solutions of the problem, as in cases I. and II. 
If the arc failed to intersect the line at all, it would prove the 
case to be impossible. 





Fig. 59. 



Fig. 60. 



The following examples will show the practical applications 
of the preceding principles : 

Example I. A crane A B C, Fig. 59, carries a load Z- at ^ ; it is of a cylindrical 
sliape at B, and held in position by a roller bearing, and at C there is also a pivot 
step. Required the forces /*! and /a <*t i? and C The centre of gravity of the crane 
itself is at 5, and its weight is equal to (?.* 

Both Z- and G act in a vertical direction, and the force at P^, if the bearing is 
smooth and we neglect its friction, acts in a horizontal direction. Combining G and 
Z. into one force Q ^= G + L, the position of whose resultant is T Q, we have the 
intersection (7 of a vertical through T Q, with a horizontal through Pi as a. point in 
the direction of the line of the force F^. This force must also act through the centre 
of the pivot C, since this is restrained from lateral motion by its bearing. This gives 
C O for the direction of the force /*.i. We can now draw the force polygon, Fig. 60, 
drawing L + G vertical, G Px parallel to O P-^, and F-i Pi parallel to O C. This 
determines the extent of both Pi Po, and by further analysis the entire load on the 
pivot C may be found. 





Fig. 61. 



Fig. 62. 



Example II. A crane constructed as shown in Fig. 61 carries a similar load to thft 
preceding. It is arranged with a cylindrical bearing at B, and at C there is a conical 
roller bearing upon a conical surface on the base of the column, the axes of both 
cones intersecting in the middle of the bearing B. 

We have, as before, the mean load Q ^ -L -\~ G ; we also have the direction of the 
pressure Pi, as it must be normal to tne surface of the cone at the point of contact. 
The intersection of /\ and Q determines O, and a line from O through the centre 
of the bearing B gives the direction of Z^- The force polygon can now be drawn, as 
shown in Fig. 62, and by making the vertical equal to Q, and the other two sides 
parallel to Pi and Pn, we determine at once the extent of the two latter forces. The 
vertical component of P^ will, in this case, be less than the total load Q, while in the 
previous example they were aJike. Hence it follows that the conical roller supports- 
a portion of the load. 





Fig. 63. 



Fig. 64. 



Example III. The crane shown in Fig. 63 is similar in construction to the pre- 
ceding one, except that the axes of the conical rollers intersect at a point D below 
the bearing B, If we now draw C O normal to the surface of contact CD, to the 
point O, and construct the force polygon, Fig. 64, we see that the change in the posi- 
tion ot the apex of the cone V causes the force /o to act from below instead of from 
above, as in the previous case. It will therefore be necessary to provide the bearing 
B with a collar to oppose the upward pressure, f 



* In ordinary wharf cranes the value of G, which mainly depends upon the capa- 
city and overhang of the crane, may be taken at ^ to J the load. 

f This defect may be seen in numerous existing examples of crane construction. 
In a case which came under the author's observation, a crane intended to have a 
capacity of thirty tons gave way under a load of only about twenty tons, because the 
proper provision was not made for the direction of a force upon a bearing. 



28 



THE CONSTRUCTOR. 



Example IV. Three forces of 70, 50 and So pounds act, as shown in Fig. 65, upon 
"a, body A B\t\ such a manner that tlieir resultant passes through the point A. Two 
■Other forces of 05 and 60 pounds also act upon the point A, and hold the preceding 
-forces in equilibrium. Required the angles which the latter forces make with the 
former. 

Lay out the forces of 70, 50 and 80 pounds, as shown from C to Z) in Fig. 66, by the 
"heavy lines, then describe from Cand D circular arcs with radii of to and 95 respec- 
tively, and thus obtain the intersections E E' or F F' , which, when juined with C 
and Z>, complete the force polygon. Both solutions are given in the diagram. 





Fig. 65. 



Fig. 66. 



Example V. An obelisk is to be raised upon its base, Fig. 69, by turning it about 
<he angle A, the lifting force to be applied in a given direction at the ape.\- B. 

Required the direction to be given to a force /^3 of given extent, applied to the 
point A, in order that the base shall only be subjected to vertical pressure. Draw a 
vertical line through the centre of gravity 6^ of the obelisk, intersecting the direction 
of the force P^ at O, A line from O through A will then give the direction of the 
resultant of the two forces. This resultant is now to be resolved into a vertical com- 
ponent Po, and a force Pj, of given extent but undetermined direction. To deter- 
inine the direction we draw, as in Fig. 68, C Q and Q P-^, and erect a perpendicular 





Fig. 67. 



Fig. 6S. 



■through /j. From Cwith a radius equivalent to Ps, describe an arc, intersecting 
the vertical at D and D' , showing that two solutions are possible— one giving P« the 
value />! D and P3, the direction DC; the other giving P„ the value P^ Z>i and' P3, 
the direction Z>i C " - 

If P3 should just equal the perpendicular distance from Cto P^ Pn, then but one 
-solution exists. The two results for the example given are shown in Fig. 67 at A Pj 
and A P3. 

Examples of this character seldom occur in actual practice. 

BQUII,IBRnjM OF InTERNAI, FORCES IN THE CORD POI,YGON. 

As already stated, we mean by the internal forces of the cord 
■or link polygon the tension or compression ■which may exist in 
the different sides of the figure, as sho^wn at 5,.j, S.,.^, etc.. Fig. 
69. These forces are of such an extent 
that they hold each other in equilibrium at 
the knots A', A', A'3, etc. Any t-wo of these, 
for example, S\.,, S.,.^, may be determined 
from their resultant P.,, when either their 
^® direction, their magnitude, or one direction 
and one magnitude are given (see ? 34). 
This is done in the following manner : 
Construct the force polygon. Fig. 70, of the 
Fig. 70. external forces A",, A, Aj, which, if equilib- 

rium exists, will form a closed figure. From 
the extremities of the sides corresponding to the force f^ 
draw two lines parallel to the sides 5,.,, S.,.^, intersecting 
at O ; then the length of the lines C, and O., will represent 
the magnitude of the stresses in the sides S^.,, S,.^. In a 
like manner we may draw lines connecting the several cor- 
ners of the polygon. Fig. 70, with the pole O, and deter- 
■mine all the internal forces of the link polygon, both in mag- 




nitude and direction ; so that when the external forces are 
known, and also the direction of two of the internal forces, 
the direction and magnitude of the others can be determined. 
This assists greatly in the construction of the link polygon, for 
by selecting one knot and determining the pole O, the sides of 
the link polygon can be drawn parallel to the respective rays. 




Fig. 69. 

The actual lengths of the sides of the link polygon are deter- 
mined by the positions of the lines of the external forces, from 
which the positions of the internal forces are also determined. 

The cord polygon will vary in its form according to the choice 
of a starting-point from which it is drawn. In Fig. 6g two 
forms are shown in dotted lines within the cross-hatched figure, 
their sides being parallel to those of the first polygon. Another 
solution of the same problem (the combination of the external 





forces into a link polygon) may be obtained by an application 
of the double solution of Case I., § 34. 

In Fig. 72 we have the directions 5]. 2 and ^'2.3 drawn from the 
extremity of the force A,, giving a new cord polj'gon, Fig. 71, 
of a verj' different form from the preceding one, which is also 
included in Fig. 71 for purposes of comparison. With the ex- 
ception of the first two sides, we have an entirely diflerent 




Fig. 71. 

figure, showing the variety of polygons which may be 1: onstructed 
from a given set of forces. 



THE CONSTRUCTOR. 



29 



The cord or link polygon, when taken in connection with the 
force polygon, forms what has been termed the graphical plan 
of forces. In most cases the entire subject can be discussed by 
the construction of one figure which may then be called the 
Force-pan, and of which examples are given in § 4S. 

Resultant of Isoi^ated Forces in One Plane. 

If we assume two of the sides of a cord polygon to be divided, 
and insert at the points of division forces corresponding to the 
stresses in the divided sides, the equilibrium will remain undis- 
turbed, as, for instance, in Fig. 73, the sides jTj A'^, and A', A'5, 
are cut and sustained. It will then be evident that the resultant 
of the forces, S^.g and St.-,, either on the right or the left will 
hold the remainder of the forces of the polygon in equilibrium. 
The position of this resultant force is determined by prolonging 
the sides until they intersect at H/. The direction and extent 
of this resultant is determined in the force polygon. Fig. 74, by 
the diagonal 4.6, which is the closing line of the forces S\.^ = 
Og, and Si3 = O4. This force is also on the one side the 
resultant of the forces /s and P^, and on the other side, of the 
forces P^, P,, jPj and P^. 

/k general it may be stated that the point of intersection of 
any two prolonged sides of the polygon is a point of the resultant 
of all the external forces beyond these sides, from which the 
direction and extent of said resultant way be determined. 

This principle is of great utilit}', as many examples will here- 
after illustrate. By reversing the above rule, the cord and force 



....,.j«....-» 




Fig. 73. 

polygons may also be used for the decomposition of forces, as 
well as their resolution. For instance, if it is desired to decom- 
pose the force 4.6 into two others, /a and i°j, of given direction, 
draw one of them (for example, Pn ) in the cord polygon until 
it intersects 4.6 in the point N, and through this point draw P5, 
parallel to the side 4.5 of the force polygon. The first chosen 
line, A'e N, may be drawn either forwards or backwards on M N, 
without disturbing the equilibrium. 

? 37- 

Conditions op Equilibrium for Isolated Forces in 

One Plane. 

In the preceding discussions it has been assumed that the 
forces whose equilibrium has been investigated were so situated 
that equilibrium really existed, so that according to the rule in 
the preceding paragraph it would be possible to reduce them to 




Fig. 75. 

two equal and opposing forces. This is, however, not neces- 
sarily the case when the force polygon is a closed figure, but it 
must follow when the cord polygon is also a closed figure, /. e., 
the actual positions of the forces must also be taken into 
account. If the positions are not correctly taken, the cord 
polygon will show what modification must be made in order to 
secure equilibrium and avoid the formation of rotating couples ; 
which will be discussed in the nest section. For this purpose 



one of the forces should lie left to be determined in position a. 
the last. 

I. Let this force be /'„, Fig. 75. Itsmaguitude is known, and 
its direction is parallel to the given line Z Z. After construct- 
ing the force polygon. Fig. 76, choose a pole O, and draw the 
rays to the angles from i to 6, so that A'l K, is parallel to I O, 
A'2 K, to 2 O, A'3 A'j to 3 O, etc., until A'5 A'o is reached. Then 
the closing line of the cord polygon must have the direction 
6 O, and must also pass through K^. This determines its posi- 
tion entirely, and its intersection A'g with A'5 A'g is a point of 
the force A'5, which is now drawn parallel to 5.6. 




Fig. 76. 




Fig. 7S. 



If the final force is not given either in direction or magnitude^ 
it may be determined from the direction and position of the 
other forces as follows : 

II. Let the yet indeterminate force be P^, Fig. 77, while wa 
have given the direction of the force P^, which is A'^ A",, and its 
position A'l. We can draw the force polygon from the points 
I to 5, while from the point i we have only given the direction 
A I. The cord polygon may also be commenced by starting 
from A'l, and continuing through the points A',, A'3, A'j, A'5 and 

■ Pi' 




Fig. 77. 
IC'^. We may then select any direction for its closing side A' L, 
and its intersection A'j with K-. A''5 will be a point in the line of 
the desired force P^. In order to determine its magnitude and 
direction, draw, in Fig. 78, O 6 parallel to A'l L, and join the 
point 5 with the point 6, when the line 5.6 will give the desired 
magnitude and direction of the force P-^. 

?38. 

Force Couples. 

When a plane figure is subjected to the action of forces in 

couples, acting in its plane in such a manner that, while equal 

in magnitude and opposite in direction, they fall upon parallel 

lines, and do not oppose each other in the same straight line. 





Fig. 79. Fig. So. 

the force polygon will close without necessarily proving the 
existence of equilibrium in the figure. 

The conditions which obtain under these circumstances may 
be examined as follows : 

The forces P^ P, and P^ P^, Fig. 79, form a closed force poly- 
gon 1,2, 3, 4, Fig. 80, but at the same time equilibrium does not 



30 



THE CONSTRUCTOR. 



exist in the figure, but instead, a tendency to rotate about a 
common point with a statical moment which is equal to the sum 
of the moments of the couples {I\ — P-^ and ^I\ — P^. In 
order to secure equilibrium it is necessary' to introduce an addi- 
tional couple (/-"s — Pf,], whose tendency shall be to cause a 
rotation in the opposite direction, and whose moment shall be 
equal to the combined moments of the previous couples, and 
whose direction shall be parallel to the lines V V and VI VI, 
Fig. 82. 

Let us take. Fig. Si, the force polygon A i, 2, 3, 4. This is 
not yet complete, for we still lack the forces 5 and 6. We know 
that they must act through A, in opposition to the other couples, 
tut their magnitude is net yet determined. 

As already said, the two forces must be equal and parallel in 
order to be in equilibrium %vith the other couples, and only two 
forces can fulfill the conditions. Their direction is given, and 
can be laid off as at A Z. We choose any pole O, and join the 
rays O A, O 'i, O z, O i, O i, (= O A), and can then proceed to 
construct the cord polygon. Fig. 82. 

For this we have lines of direction //, Till, etc., up to VI 
VI, given from Fig. 79. Starting from any point K^ on //, we 
draw lines parallel to the rays O A and O 1 (their resultant 
being the force P^] until they intersect VI VI in K^, and II II 
in A', ; then draw A', A', parallel to O A, intersecting III III at 
A'3, A'3 K^ parallel to (9 3 until it intersects IV II', and K^ K^ 
parallel to O 4, intersecting V V at A'5. Onl3- the closing line 




I 

E, 

5- n 


/ 


My 


X 

m 


Ka 


v/\ 'y 




Q 


y K4, 



Fig. 81. 



Fig. 82. 



of the cord polygon is now lacking, as 't is the line joining K^ 
with A'5. which latter point has already been determined. We 
can now (see J 37, II.) draw the ray O, parallel to A'j A'g, com^ 
pleting the force polygon and the line A 5, will give the magni- 
tudes of P^ and I\. The path around the force polygon may 
be taken as A i, 2, 3, 4, 5 A, the sides 4.5 and 5 A being sup- 
posed to make an infinitely small angle with each other. 

The previous examples upon the force and cord polygon serve 
to show how geometrical addition and subtraction may be used 
to determine the equilibrium of diverging forces in one plane. 
Forces acting in intersecting or parallel planes may be examined 
in the same manner, and in many cases without a great degree 
of complication, as some following examples will illustrate. It 
is not intended, however, to undertake a general discussion of 
the subject here, but rather proceed at once to practical appli- 
cations of the special case of parallel forces. 

I 39- 

Equilibrium Between Three Parai,i,el Forces. 

In discussing the equilibrium between parallel forces, we may 

use purely arithmetical methods, or use geometrical addition 

and subtraction (force and cord polygons), as may be found 

most convenient. 

The present problem may be stated as that in which a force Q 
acts upon a body, and is to be held in equilibrium by two un- 



P. 




Fig. S3. Fig. 84. 

tnown forces, P, and P„, acting parallel to it and to each other. 
Drawing the line ABC, Fig. S3, normal to the given direc- 
tion of the forces, we must have, in the existence of equilibrium, 
J'l . Ali = P. . B~C, or P, rti = Pj a^, and also Ai -f Pj = Q. 



P, — ~ graphically, we may follow 



In order to determine A\ 
tie method in ? 24, and in Fig. S4"make O E ^ the divisor a^ 



O A = the factor a,, and taking E P to represent temporarily 
the force P.^, draw A C parallel to E B, which gives the propor- 
tional value of A,. By placing the triangle C A O in the dotted 
position O' B A', we have A' E = I\\ P^= O. This gives 
a figure in a form well suited for application to Fig. S3, as will 
be shown in the following examples : 

I. In Fig. 85 draw A D equal in value to O, join D with the 
third point of application C, and prolong Q until it intersects 




Fig. 85. 



Fig. 86. 



at A" a line drawn through A? parallel to A C. Then will we 
have the following relations, B E= P^,E F= P,. In Fig. 86 
is shown a similar case, but with O inclined to A B C, and in 
Fig. 87 O is beyond AC 

II. By resolving the force O into two components applied at 
the points A and C, Figs. 88, 89, 90, we obtain inclined forces 




I^G. 88. Fig. 89. Fig. 90. 

■whose components parallel to O are the desired values for Pj 
and P.,, while the components which are parallel to A B C neu- 
tralize each other. In all three figures B F= A, and F D = A,. 
III. By constructing the force pol3'gon, making A D ^ Q, 
and using any pole O, Figs. 91, 92, 93, and drawing the sides of 
the link polygon, so that A b\s, parallel to A O, b c parallel to 




Fig. 91. 



Fig. 92. 



Fig. 93. 



D O, and joining the closing line c A, the parallel to the latter 
in the force polygon O E will give E A = P^, and D E ^ P„. 

If it is desired to make the closing line fall upon A B C, or 
lie parallel to it, the cord polygon A. b Cmust be first drawn, 
and the pole O, determined by the intersection with A B oi 3. 
line D O parallel to b C, P> A having first been drawn equal to 
Q, O E may then be drawn parallel to A b, and we have 
E A = P^, and E D = P.,. _ 

In these cases Q is equal in magnitude to the resultant of P^ 
and P.,, and opposed to them in direction. If Q is to be deter- 
mined when Aj and P^ are given, similar methods to the fore- 
going are to be followed. 

Returning to the diagram O E A C B, Fig. 94, which we have 
already used in case A, we construct the triangles C A O and 











9' 


^ 


B 


c 










-i 




Fig. 94. Fig. 95. Fig. 96. 

B A' O', and draw B' C parallel to O A ; O' C and O B' par- 
allel to A' B, giving B' B = a^, B C = a^, B' O = P.„ O' C 

= ^1- . . 

From this we obtain the following solutions : 

IV. Transfer one of the forces to the opposite side of A C, 




P 
Fig. 97. Fig. 

Figs. 95, 96, so that A D =^ P.^ and E C= Pi, join D to E, and 




THE CONSTRUCTOR. 



31 



the line D E will intersect A Cat B, vchich -will be the point 
of application of the resultant Q, whose magnitude = £ D' 
-= P^-\- P^, since D D' is drawn parallel to A C- 

In Fig. 96 jP, and P., act in opposite directions, and their alge- 
braic sum D' E must be taken, and, as shown, the resultant Q 
acts bej'ond A C. 

V. The method shown in Fig. 97 follows from (II) : 

From the extremity a oi a A ■=■ P^ draw a line A' a of any 
length, making it parallel to A C. In a similar manner draw 
c C from the extremity oi c C =^ P.,. Draw A' A and C C, 
prolonging them until they meet at E, which latler will be a 
point in the line of the resultant E £, and the value of Q 
will be P-i + P^, which is also the resultant ol D E =^ C C and 
EE=A' A. 

VI. Following the method in (III), we may proceed as fol- 
lows, Fig. 98 : Make D E = P„, E A = P^, choose a pole O, 
and join the c'osing line O E oi the force polygon. Draw A c 
parallel to E O, c b parallel to O D, and A b parallel to (or, as 
in this case, the prolongation of) A O, and the intersection b 
will be a point in the line of the resultant O, whose maenitude 
= jyA. 

?4o. 

Resui<tant of Several Parallel Forces. 

When we have a number of parallel forces Qi, Q^, Qi, Qt, 
acting upon a body in given positions in one plane, we can 
determine their resultant bj' a combination of the preceding 
methods, resolving them in pairs until all are combined. 



Qi 



JQ2 



,''!> 



Qi 



-'] 
I 
i 

-jfl' 



"E 



Fig. 99. 

I. In order to combine the forces Qi to Qi, intersecting a 
common normal A E, Fig. 99, we first combine O^ and O. by 
transposition, as in Fig. 96, and obtain the resultant, Qi + O2 
= b c. This may then be combined with O^. giving ^t^' = 
!2i + ft + Qi< snd this result with Qi, which finally gives the 
resultant, 5= !3i + {?2+ ft+i passing through 3T. This solu- 
tion is one which is sometimes desirable in machine construc- 
tion, as, for example, in the distribution of the weight of a loco- 
motive engine vipon the various axles. The method of deter- 
mining the resultant of several parallel forces in this way by 
the successive combination of pairs is very tedious and of limited 
application, and the method given below of using the force and 
cord polygons is much simpler. 




Fig. 100. 

II. Fig. 100. Form the force polygon of the given forces ft 
to Of,, by laying off lines successively from A, equal in length 
to the magnitudes of the several forces A i, 2. 3, 4, 5, 6, as 
shown in the left of the figure. The magnitude of the resultant 
will then be equal to the length of the closing line 6 A. 

To determine its position, proceed as follows : Select any 
point beside the line A s, as a. pole O. and join the rays O A, 
Ol, O 2, O 3, etc. Starting from a point b under Oi, draw b b' 
parallel to A O, and b c parallel to i C, and continue by drawing 



r^ parallel to 2 O, d e parallel to 3 O, etc., and finally reaching 
the closing line of the pol3'gon g g' parallel to O 6, intersecting 
b b' at q, which determines the position of the resultant Q (see 
§35)- 

The method shown in (J 36) may also easily be applied to the 
resolution of such forces, as in Fig. 100, the intersection of d c, 
prolonged to c' , gives the position of the resultant of O^ and ft, 
and its magnitude is shown at ^4 . ., in the force polygon, and in 
a like manner c' is the position of the resultant of £?4 and Q^. 

Decomposition of Forces into Two or More Parallel 
Forces. 

The methods of resolving forces by means of the cord poly- 
gon will also serve for their decomposition. 

If, for example, in anj' portion of a cord polj-gon a q b c d. 
Fig. loi, it is desired to substitute for a force Q, two forces Q-^ 
and Q., passing through e and/", we have oulj- to join the points 
e and f to obtain the form of cord polygon for the new forces. 




Fig. 101. 



Fig. 102. 



and determine their relative magnitudes by drawing 0\ parallel 
to ^y in the force polygon below. If the required force O-^ and 
(Jo both lie on the same side of O, Fig. 102, the solution is 
similar. We now prolong a q to its intersection e with O^, and 
join e f. Also mark the intersection of (2i with q b, and O, with 
q a. In the force polygon below we have Q-^=^ A \, 0.,^= \ . 2, 
or A i' = Q^anAV '2 = O^. 

If we have a beam A G loaded with parallel forces Q^ to ft, 




Fig. 103. 

Fig. 103, whose load is to be opposed b)' reactions P^ and P^ at 
A and G, we may first determine a resultant f) of all the forces, 
as in [f. 40), and then decompose this into values for Z', and P, 
by the method just given. We also omit the determination of 
O altogether,' and proceed to determine P^ and P^ directlj' as 
follows : 

Choose an}' pole O, and form the force polygon .A'l . 2 . . . . 5 ft 
and construct the cord polygon, making its sides parallel to 
their respective ra5'S, and draw b a parallel to A' O and/g, par- 
allel to O 5, their intersections Viiith the lines of the forces P^ 
and P, beings and 0-. Join ag, which will be the closing line of 
the polygon, and its parallel 06 in the force polj'gon gives P., = 
5. 6 and /"j =: 6 . 7. If the sides a b and/g of the cord pol3-gon 
are prolonged in the other direction we obtain a' and g', giving, 
however, the same result, since n' g' is parallel to ag. The 
cord pol5-gon would then be the figure a' g' m b d c efiii a', and 
111 indicates the position of the resultant of the forces Q^ to ft, 
or of Pi and P„. 

■When a loaded beam is supported by three or more bearings 
it is necessary to take into account the resistance of the beam 
itself with some degree of accuracy, or else the problem be- 
comes indeterminate. This indeterminate character may, how- 
ever, be eliminated by the introduction of an equalizing lever. 



THE CONSTRUCTOR. 



Suppose we have, Fig. 104, a beam BCD, the resultant Q of 
whose entire load acts at M, and is opposed by the reactions of 
three supports at /"i, P.,, P^, at right angles through the points 
B, CanAB. 

We may now assume, temporarily, an approximate ratio be- 
tween two of the forces, e.g., P^ and Pn, and permit them to act 
at the extremities of an equalizing beam B^ C^, which in turn 
supports the main beam at ^ £■! ; makingthe ratio of ^1 C^ :E^By 
the same as has been chosen for P^ : P.,_. Now decompose Q into 
the components acting at E and D by means of the cord and 

*^3 




Fig. 104. 

force polygons e m d and A O \ 2. This gives y4 1 = 5, i . 2 = 
P^, 2 Ji =: Pi-\- P.,, which last sum may be then divided between 
Bi and Ci by any of the above methods. p 

Each different approximation of the ratio -J, will give a dif- 

ferent value for P^. If P, and P^ are made equal to each other, 
£■ will be in the middle of B C, and the equalizing .lever will be 
of equal arms. The distribution of the load of locomotives and 
cars upon their spring is usually made with such equalizing 
levers. 

If the load is to be supported upon more than three or four 
points it will be necessary to use several equalizing levers, and 




Fig. 105. 

examples of this will be found in some locomotives. If, for 
example, we suppose M, Fig. 105, to be the point of application 
of the total load ^ of a locomotive, supported upon three axles 
B CD in such a manner that the weight shall be transmitted to 
the axles through the springs as shown, and also that the 
■weights upon the wheels Caud D shall bear a determinate rela- 
tion to each other. This can be accomplished by the use of 
three springs and one equalizing lever upon each frame of the 
locomotive, the whole weight being thus supported upon eight 
points. Taking the relation between the forces P, and P-^^p : q, 
we erect a perpendicular E e, whose distance from the axle C 
and i? is in the proportion q : p. From any point e' on this 




for the equalizing lever. The distances of the points c and d 
from the verticals through Cand D give the length of the arms 
of the springs fj c.^ and d^ d^. These springs are made with 
arms of equal stiffness, since they are to support equal loads at 
both ends. For any chosen ratio p : q, and given distance be- 
tween the axles, the actual length of the equalizing lever will 
not affect the ratio of P^ to the sum P^ ■\- P^, as an inspection 
of the cord polygon b m c d will show. 

The springs which are attached to the ends of the equalizing 
lever must, of course, be made of sufficient stiffness to support 
the load which is thrown upon them, and the length of the sup- 
ports and their proportions chosen according to the previously 
determined distribution of the weight. 

Many similar examples to the preceding might be given, as 
they are of frequent occurrence in practice. The two springs 
which are attached to the equalizing lever may be replaced by a. 
single spring, as in Fig. 106. In this case the axes C C are con- 
nected rigidly to the lever dec, and the lever itself rests upon 
a spring b^ e^ c^, whose extremities are fastened to the frame. 

The arms b^ e^ and c^ e^ of the spring are of unequal length, 
and have the same relation p : q as that which exists betweeiL 
the arms of the lever bee. If the arms of the lever are not 
properly proportioned, or if any error has been made in the dis- 
tribution of the load, it will be made apparent by the inclined 
position which will be assumed by the equalizing lever. 

?42. 

Uniformly Distributed Parallel Forces. 

When a beam is subjected to a uniformly distributed load, the 

force and cord polygons cannot be determined by the preceding 

methods, since in such cases the cord polygon becomes a figure 



Fig. 106. 



line draw lines to the bearing points of the wheels upon the 
rails, and any horizontal line will intersect these inclosed lines 
in points which will give the proportional length of arms c e d 



. I 2 

1 r 


3 


r r 


6 


7 


8 9 










E 










n 


\ 


,b 

C 


< 


X 


^ 


t-^^ 


h 


/ 





111 


;| 


i 






















■R 


— 






- 


t: 


-" 




'J 


7 


\ 


\ 


^ 


X 


1 


f 


V 


V 







it 

Fig. 107. 



H. 

Fig. 108. 



of curved outline. The character of the curve may be deter- 
mined in the following manner : If we assume the load to be 

concentrated at a number of equidistant points, as in i, 2, 9, 

Fig. 107, and construct the cord polygon for these conditions, it 
will be evident that the sides a 7)/ and b c will intersect midway 
between i a and 2 b, and also midway between a b' , since the 
forces I and 2 are equal to each other. In the same manner cd 
and a i)/ intersect midway between 3 c and i a, which is also in 
the line of 2 b, that is, at b' , and likewise d e and a yJ/ intersect 
midway between b' and c' . In this way it may be shown that 
the intersections of the prolonged sides of the polygon from 
a Jl/ to iM are at equal distances from each other. This indi- 
cates a known property of the parabola whose vertex lies on, 

E J>/ 
line E M, and whose abscissa e E ^ . This parabola is 

the form assumed by the cord polygon when the load is uni- 
formly distributed, as was previously assumed. If we note that 
the triangle A M B represents the entire load collected at E, it 
"will readily be seen how the curve may be drawn in any case. 
If the chord A E B is inclined, as shown in Fig. loS, the divi- 
sions of A 77/ and MB will be equal in number, but the divi- 
sions of A /)/will be of different size from those of 71/ B. The 
point e lies in the middle of E M, but is not the vertex of the 
parabola. 

Link polygons which assume the form of curves ma}' also be 
used to show the effect of moving loads, and are then the figures 
which are contained within the successive sides of a regular 
polygon. Many examples are to be found in the case of railway 
bridges, traveling cranes, engine guide bars, etc. 

§ 43- 

The investigation of the action of parallel forces, such as Qj 
to Qi and /\ P.,, Fig. 109, whose direction is normal to a beam, 
requires a knowledge of the statical moments of the external 
forces. These can best be obtained by use of the force and cord 
polygons. After constructing the force polygon A O /^, and cord 
polygon abed ef, let it be required to find the statical moment 
for any point .S upon the beam. This moment is the product of 
the resultant of all the forces upon one side or the other of the 
line 5' S^ into the lever arm / of this resultant from 6" .S,. 

The magnitude of this resultant is obtained from the distance 
A J ^ 1 , 5 in the force polygon, cut off by the rays O l and O 5, 



THE CONSTRUCTOR. 



33 



which are parallel to b c and/ 3, and its point of application is 
determined by prolonging these sides until they intersect at g. 
By drawing the perpendicular ^ o-^.,the lever arm I of the re- 
sultant Pz= h i is determined, for the force acting at the point 
S, and hence we have 7)/ = P I. 

This multiplication may also be performed graphically. By 
drawing the perpendicular O k va. the force polj-gon, we obtain 
the altitude of the triangle O h i from the base h i, and this tri- 
angle is similar to the triangle g s So, whose altitude is /. Call 
in O k = H and s Sa ^ t, we hava 

P:H=t:l, 
or M=Pl = Ht. 

This proves that the statical moment at any point in a beam is 
proportional to the corresponding ordinate of the cord polygon, 
parallel to the direction of the forces, since // is a constant. 
By making H equal to unity the conditions become similar to 
Case I., ? 22, in graphical multiplication, and the moment M 
becomes equal to the ordinate t. It is not necessar}' to deter- 
mine the position of the point of application^ of the resultant, 




since it is the relation between the statical moments which is 
most desirable, whether //"be chosen as a unit or not. This 
property of the cord polygon for parallel forces is most useful, 
and an example may be found in the case of axles. 

For such cases as for many others, it is most useful, since no 
modification of the diagram is necessary, the moments being 
found by the same construction which is required for the deter- 
mination of the forces. It is often convenient in practice to 
cover the figure containing the moment ordinates with section 
lining or with a light tint of color. 

?44. 

Composition and Decomposition op Staticai, Moments. 

As shown in the preceding section, statical moments may be 
shown by means of lines of definite length and position in the 
same manner as simple forces. When two statical moments act 
in the same or in different directions, they may be combined by 
means of graphical addition in the same manner as has already 




been shown in § 31. li A B Cand ADC, Fig. no, represent 
the cord polygons for two sets of parallel forces which act nor- 
mal to the axis of a revolving body A C, in the directions A' B' 
and A' D' we have the following method : For a point ^ on the 
axis of the body we have the triangle T-^ S T/, in which the 
angle <t, = B' A' D' and T-, T,' = S T/ = t for the desired 
moment. The combination of the cord polygons ABC and 
ADC, which may be called the moment surfaces, will give the 
resultant moment surface A T U C. The sides A 7 and C U 
are here straight lines, while T U' \s, a. curve, in most cases a 
hyperbola. In actual practice the straight line joining Tand U 
may be used with but little error, and its detailed construction 
is unnecessary. 

By a reversal of the above construction it is possible to decom- 
pose any given statical moment t into two others, t^ and t^, if 
their directions be given. 



?4S. 
Twisting Moments and their Graphicai< Combination 
WITH Bending Moments. 
Next in importance to bending moments, and often acting in 
combination with them, are twisting moments. In Fig. iii let 
A B CD be the axis of a rotating body, subjected to bending 
forces at B C, and supported at A and D ; the force polygon 
being represented at A O -2. and the moment-surface at A 6 c D, 
and let the portion between B and Cbe subjected to a twisting 
moment P . R, and the moment-surface of the latter be required. 




Fig. III. 

According to \ 43, and the method of multiplication given In 
Rule I., ^ 22, we find a line corresponding in value \.o P Rhy 
laj'iug off in the force polygon A p ^ P, joining the ray Op, 
prolonging O A and Op, and drawing q r parallel to A p at a 
distance equal to R, giving a length q r equal to P /'.upon the 
same scale used for the polygon A b cD. The moment-surface 
for the twisting between B and (Twill then be included in the 
rectangle B Cv u, whose altitude B u = C v ^ q r. In common 
practice it is desirable to convert this torsion surface into one 
representing equivalent bending moment. This may be done 
by taking a proportional value which shall give the same 
security as the bending moment. It has been shown in § iS 

that the latter is equal to —^ the twisting moment. We may 
o 

them make B u^ ^ C Vi = -5- B u, in order to obtain the mo- 
o 

ment-surface of the bending moment between B and C, which 
may be measured upon the same scale as A b c d. 

If we wish to combine this with the given bending moment 
we may do so graphically by first using the formula IV. of the 
table of ?. 18, p. 60, in which the ideal bending moment for the 
combined action of a twisting moment 3Id and a bending mo- 
ment 3ft is : 

5 



Mi 



Mb -f 



/■ 



Mb' 



In this case we make B b^= -^ B b, C c^ ■■ 



Md" 
S 



C c, E e 



~ E e, etc., rotate B ii-^^, C v-^ and E w-^, down upon A D, and 

o 

add the hypoteneuses b^ n^', c^ z//, e^ w^' to the lines b b^, c c^, e e^. 
The combined length of these lines gives the length for the 
ordinates at ^, Cand D, from which the resulting ideal cord 
polygon B bb' e' c' c D may be constructed. 

§46. 

Determination op the Centre op Gravity by means 

of the Force Plan. 

The position of the centre of gravity of a plane figure may 

I 




often be very conveniently determined by means of the force 
plan. This may be done by dividing the figure into a number 
of strips of uniform width, such that their area may be con- 
sidered as proportional to their middle ordinate, constructing 



34 



THE CONSTRUCTOR. 



the force and cord polygons, and taking the line of the resultant 
as a line of gravity. If the figure is not symmetrical, it will be 
necessary to divide the figure again in another direction and 
determine another line of gravity, when the position of the 
centre of gravity will be found at the intersection of the two 
lines. For figures of simple form larger determinate sections 
may be taken instead of strips, their area determined in any 
convenient manner, and the diagram constructed accordingly. 

Suppose, for example, that it is required to determine the 
position of the centre of gravity of the T-shaped section shown 
in Fig. 112. The figure is symmetrical about the axis Y Y, so 
that the centre of gravity must lie somewhere in that line. We 
may divide the figure into the rectangular b X c, bi X i^i and 
^2 X £", which we will call respectively the areas i, 2 and 3. 

We have also given c ^ i . 5 62 and c-^ = b.^. This gives the 

three forces as i . =; — , -^—, and —^, which are then laid off at 
"^2 2 2 

^ I 2 3, a pole O selected, and K-/ K^ drawn parallel to O A, 

K-i K^ parallel to O I, K.^ K^ parallel to O 2, A'- Ay parallel to 

O 3, when the intersection of the sides A', A'/ and Afj Ay at M 

gives a point on the line of gravit}' 71/ M' , whose intersection 

S with the axis Y Y is the centre of gravity of the figure. 

§47. 
RESULTANT OP THE Load on a Water Wheel. 
It is very important in designing a water wheel to be able to 
determine the position of the resultant of the water acting upon 
it, and the method of doing so will furnish an excellent illus- 
tration of the application of the principles of the preceding 
sections. 




Fig. 113. 

In the breast wheel, which is shown in Fig. 113, there are ten 
buckets in the half section, the third from the top being the 
first to receive a charge of water, the amount being estimated 
from a previously determined coefficient. The level of the 
water in the succeeding buckets may be considered as horizon- 
tal, and the discharge from the buckets is prevented by the cul- 
vert AT/,, so that if we neglect the leakage around the edges of 
the culvert we may count that all the buckets from No. III. to 
No. X. contain the same load of water, acting in each case as if 
its weight were concentrated at the centre of gravity of each of 



the respective prisms of water. Bucket No, XI. we may con- 
sider as entirely empty. 

I. Determination of the culvert arc K L. The contents of a 
bucket section are determined by the cross section contained 
between two adjoining bucket divisions prolonged, as governed 
by the coefficient of charge, = 0.04. Now, in bucket I. lay off 
k I = 0.4 of the bucket spacing, and draw / in radial ; then the 
section k I in n will represent a bucket charge. In bucket II. 
its figure assumes the shape r p u 1, in which the angle t is the 
beginning of the scoop of the bucket r t, and /!■ / u will be equal 
to the desired culvert angle A' 31 L, and // / J\I will be equal to 
the complement yV M K. 

The construction is as follows : In the right angled triangle 
o p q make o p = the middle breadth of the figure k I m «, and 
also make p q ^ 2 . I in \ then transform this triangle into one of 
equal area, rp s (by drawing o s parallel to rq, and joining rs, 
see 'i 25. Join j i, and draw r u parallel to it, and join u and t, 
and if we neglect the curvature of p u, we may consider the 
quadrilateral r p u t as the form of a filled bucket just at the 
moment of discharge. This requires the angle K 31 N= u 1 3T, 
but owing to the splashing of the water, the culvert is raised as 
high asy. 

II. Determination of the water level in the various buckets. 
We will begin with bucket IV. Here the figure r p t ii, is again 
drawn, and the line i' v, and its parallel w it', determined ex- 
perimentally, so that the diagonal w v shall be horizontal, which 
may readily be done after a few trials. Proceed in the same 
manner with buckets V., VI. and VII. 

In bucket III. the figure r p u t is first converted into the 
quadrilateral with the upper line p x, and this then into the 
pentagonal figure, with the level upper line j' z. 

In bucket VIII. we first get the figure with the upper line/iA-j, 
and then from this the figure with the level upper line j'j z-^. 
Proceed in the same manner for buckets IX. and X. 

III. Force plan for the water load. Now determine the cen- 
tre of gravity for each loaded bucket, and also lay off the force 
polygon A O '& for these eight forces. From this constant the 
link polj'gon d b e afg h i, according to the methods previously 
given, and z, will be a point in the resultant of all the forces. 
It is to be noticed that the centres C and JJ fall so nearly in the 
same line that their forces have been united, so that i d is par- 
allel to A O, d b to O 2, and the intermediate parallel to O i 
omitted. 

Suppose, now, that it is desired to determine completely the 
position of the centre of gravity P, of all the prisms of water. 
Draw through A, B, C, etc., horizontal lines, assume the force 
polygon A OS, to be turned around 90°, and draw a second cord 
polygon ; or, what is shorter, draw the second polygon with its 
sides normal to the rays of the force polygon, giving the figure 
a'b'c'd'e'f '---?''. A horizontal through i will then intersect 
the vertical which was previously determined, and so fix the 
position P, of the centre of gravity of the entire mass of water. 
By taking the buckets in a different position, a slight difference 
in the position of i P, may be found ; but in most cases the 
deviation will be very slight. 

?48. 

Force Plans for Framed Structures. 

Framed structures are of very general application wherever 
loads are to be supported, and their discussion may be classified 
as a system by itself, while their use extends from the simple 
trussed beam to the bridge and roof truss ; also for walking 
beams and many other uses. 

The tensile and compressive stresses iu these various forms 
may readily be examined by means of the force plan, which 
consists of both the force and cord polygons and their modifica- 
tions. The subsequent examples will serve to illustrate the 
principal cases. In all of these cases it is assumed that at the 
knots — z'.if., at the points where several members meet,— a joint 
is supposed to exist ; or at least no account is taken of the re- 
sistance to bending at the knots. 

In order to form such a plan for any given construction, it is 
necessary first to determine the division and direction of the 
forces, and then, beginning at one of the external forces and 
laying off its direction and magnitude to the next knot, com- 
bining it there with the external forces at that point, la}'ing off 
the resultant to the next bend, etc. Upon such combinations 
of force triangles or quadrangles the force plan is constructed. 

If it is desired to determine the directions of the components 
of a given or determined force, the principles laid down in | 32 
must be borne in mind. These may be generally expressed in 
the following rules : 

If one force is to be separated into two or more forces, its di- 
rection is to be reversed and it is to be made the dosing line 
S' in the patlis of the ot/ier forces. 

If two or more given forces are to be conbined with two or 
more other forces, the force polygon will consist of the given 
forces and their closing line S. 



777^ CONSTRUCTOR. 



35 



The first rule is only a special case under the second or 
general rule, since the single force may be considered as an un- 
closed force pol3'gou whose closing line passes back-ward over 
the same path to the starting point. 





Fig. 114. 



Fig. 115. 



In the investigation of each member in a frame without error, 
it is best to assume the member to be cut, and to determine the 
external forces at each section which oppose the internal forces ; 
the direction of the forces may then also be determined with 
precision. 

?49- 
Force Plans for Fr.^med Structures. 

I. Simple Trussed Beams. Fig. 11 5. The beam A B C i?. 
supposed to carry at j? a load equal to 2 P, acting in a direction 
normal to A C, and to be supported at A and C. Since A B 
= B C, the reaction at each support is equal to P- It is then 
required to determine the stresses upon the various members 
from I to 5, as marked in the figure. 

Referring to the diagram marked a, let a i be the reaction P, 
-which acts upward at A. We now have to construct a diagram 




Fig, II 5. 

of the internai lorces acting xnAB and A D. To simplify mat- 
ters, we will give these forces the same numbers as their corre- 
sponding members ; drawing 1 parallel to A B, and 2 parallel 
to A D. The direction of the force P, in the closing line of the 
force-triangle, determines the direction in the other two sides, as 
shown by the arrows, by the lines i and 2. (See ^ 48.) In this 
case there will be compression in A B and tension in ^ Z*. 

In order to show this clearly, in all the following strain dia- 
grams the forces acting compressively in struts or pests will be 
indicated by double lines, while all tension member.), links or 
rods will be shown by single lines. ■** 

Following out this idea, we shall, in the following illustrations, 
show all struts or compression members in the construction 
drawings as having a measurable thickness, as if made of wood, 
while the tension members will be represented by simple lines, 
although this is not intended to indicate any limit as to the 
choice of materials. 

For the knot at B we make a b c= 2 P, and, following in the 
direction d a t: (because the thrust is from A towards B), and 
join the closing lines 3 and 4, both of which represent compres- 
sion. The combination of 2 and 3 determines 5, which is ten- 
sion. This gives an entirelj^ symmetrical plan, which was to be 






I 


p 

b 






ftf 


3 


K 


> 


\ 


p. 




5 


»i 




X 





Fig. 117. 

expected from the symmetrical form of the structure, and an 

investigation of one-half is practically sufScient. 

If the load 2 P is taken as uniformly distributed over the 

entire distance ABC, instead of being concentrated at B, the 

P 
reactions at A and B will each be equal to — , and the load at 

i? = /*, so that )^ of the load on.ABa.n.d.B C" is referred to the 
knots A, B and C. From these conditions we obtain the force 
plan b, which is geometrically similar to the other, but only half 
as large. 



II. Double-trussed Beam (much used for constructions of all 
sizes). Fig. 117. In this case take vertical forces P, at B and 
C, and corresponding vertical reactions at A and D. In the 
first force plan, a is drawn equal to P, and I and 2 parallel re- 
spectively to A B and A E, thus determining the forces i and 
2; I, being compression and 2, tension. Lines now drawn par- 
allel to B £ and £ P, determine the compression in 3, and the 
tension in 5, while the compression at 4 is the closing line of 
3, l,andP; and the other half of the diagram is similar. If the 
vertical forces at A and B are not of the same magnitude, 
which is often the case in practice, the structure should be 
strengthened by the introduction of the diagonals £ C and 
B P. 

The second diagram shows the construction in this case. Let 
Pi = Hi bi be the force acting at A, and Pj = "2 ''■2 ^t B. Draw 
a vertical line from I to a horizontal through Q, which gives 
the length 3, of the vertical force at B, and by drawing the 



3P 

2 

A ^ L - 


P 2P 
B 4 C 


8P 
2P 

8 D 12 E 




F 




7 


^1o 





^ 


f^*^- 


- — — -: 




8^ 


><^^ 



Fig. 118. 

dotted diagonal line their resultant is found. If any of the ten- 
sion members are omitted the framework will tend to take an in- 
clined position until the various parts are at such an angle with 
each other that both constructions will give the same value for 
3. For this reason it is best in nearly every case to use the di- 
agonal counterbraces. 

III. Triple Trussed Beam. Fig. 118. The uniformly distrib- 
uted load upon the framework gives the following distribution of 
forces. The force 3 P =: a 5 c is first decomposed in 2 and i, 
OT c e and e a ; then i, is connected toa 6 = 2 P, hy the line b e, 
and this latter decomposed into 3 and 4 or ^ 7^ and J^ b ; 2 
and 3 are now joined by /" c, and the components at 5 and 6 
or /£■ and o- c found. Since 5 and 10 are equal to each other, 
we may draw c h parallel to G H, and equal to c;^, which gives 
g h ^j \ the rest of the force plan is similar to the first half. 





^' This distinction has been suggested by Cuhnann. 



Fig. 119. 



IV. " Another form of Triple Trussed Beam is shown in Fig. 
119. The space between j? and C" is twice as great as between A 
and B, and the uniformly distributed load is equal to 12 P, act- 
ing at the various knots as shown in the figure. 

In the force plan, make a b c =^ $ P, and draw parallel to i 
and 2, the lines a e and e c ; then join, i with j,p (for the knot 
at B), and decompose into 3 and 4, or e _/ and f b. Now com- 
bine 2 with 3, giving c f, and draw 5 and 6 parallel to P (Tand 
F G, respectively. This case differs from the preceding, in that 
5 is now compression instead of tension. The equality of the 
forces 6 and 10 gives g h = J, and the similar half of the dia- 
gram need not be drawn. 

V. Multiple Trussed Beam. Fig. 120. The beam A J is 
divided into eight equal parts, which are represented as being 
uniformly loaded, the load at each knot being shown in the 
figure. In constructing the force plan we make a e ^= j P, and 
by drawing the lines parallel to i and 2, we obtain a f anA f e ; 
then lay off a b =^ 2 P, and join the resultant bf. This decom- 
poses into 3 and 4, or fg and^ b. The forces 2 and 3 combine 
to give the resultant g e, which, by drawing lines parallel to 
K Cand K L, gives g h and k e for the values of 5 and 6. We 
now find that to proceed further we have three forces of given 
direction onlj', and since this is indeterminate, we must obtain 
one magnitude as well. This, for example, may be done for 
the force 7 as follows : The strut C L sustains the vertical com- 
ponents of 5 and 9, as well as its own direct load 2 P. Now 5 
and 9 are equal to each other, since they are placed sj'mmetri- 
cally, and carry equal loads from the struts i? A' and KM, 
Hence in the force plan we maj' make h i, which represents the 



36 



THE CONSTRUCTOR. 



force 7, equal to twice the projection of 5 upon tlie vertical 
■\- 1. P. This we can now combine with 6 = // e, giving i e, 
which in turn decomposes into i in and m c, or 10 and 11. 
Returning to the knot C, we may now take the line // /, and by 
drawing parallels to CL, CM and CD, obtain the figure h i k c. 





Fig. 120. 

■which determines the forces S and 9. In the same manner pro- 
ceed from 12 to 15, which will complete the half plan. It may 
be noted that the principal beam A J is subjected to a uniform 
compression throughout its entire length. 

The force plan will, of course, be modified by various dis- 
tributions of the load, as in the case of simple beams, as shown 
in cases XII. and XIII., ? 6. 

? 5°- 
Force Pl.^ns for Roop Trusses. 
Roof trusses furnish many and varied examples of frame- 
work.* In the following examples a uniformly distributed 





Fig. 121. 

vertical load is assumed, so that the burden upon any portion 
of a rafter may be considered as proportional to the length of 
that portion. 

I. Roof with Simple Principals. Fig. 121. A uniform load 
2 j°upon each half gives as the external forces P, 2 /"and P a.t 
A, B and C. Lay off in the force plan a b =^ P, and draw a c 
and b c parallel to A B and A C, determining the forces i and 
2 ; I being compression and 2 tension. Then dravi' the vertical 
c e, and also draw b e parallel to CD. thus giving both 3 and 5, 
and the diagram is completed by drawing d e. 





resultant into d e and e b respectively parallel X.o E C and E .S^. 
giving the forces 3 and 4, both being compression. By repeat- 
ing 2 and 3, in drawing 7 and S, we obtain the figure c d e f g, 
in which eg gives 5. This latter force might also have beett 



Fig. 122. 

II. Roof with Single-Trussed Principals. Fig. 122. This 
form is similar to the preceding, with the addition of the struts 
C E and C F. The distance ^ i? is to E B, as 2, is to 2 ; and 
the loads upon the respective portions are 6 /'and H P, which 
give the forces at the various knots as shown in the figure. 
Make a c \n the force plan equal to 7 P, and by drawing lines 
parallel to A E and A C, obtain the forces i and 2, or a d and 
d c ; then combine i with 5 P^ « 5, and decompose the dotted 

'^■- Many subjects for Graphical Analysis may be found in Ritter's " Roof 
and Bridge Construction," Hannover, 1863, in which the forces in the vari- 
ous members will also be found carefully determined numerically, thus 
affording convenient proof. 





Fig. 



123. 



found by combining 4 and 4 P, and decomposing the resultant 
by lines parallel \.o B C and B F, an illustration of the various 
methods in which the force plan may be used. 

III. Another form, with Single Trussed Principals. Fig. 123. 
This roof is similar to the preceding except that the struts E 
Cand C F are placed horizontally. In this case A E ^ E B^ 
and the external forces at A and D are both equal to $ P. 

2P 




Fig. 124. 

The forces from a to ^ in the force plan are determined as be- 
fore, giving da and cd for the forces i and 2, and the com- 
bination of I with 2 P gives the resultant d b, from which the 
thrusts 3 and 4, or d e and e b, are obtained. The value of i 
is the same as 3, and 8 is the same as 2 ; while 5 is the closing; 
line oi c d e d f, or of c df. The force 5 must also be the com- 
bination of the equal forces 4 and 6 with 2 P, which the dia- 
gram shows to be the case. If the rod C B is omitted, as is- 
frecpiently done, the strut E C F, if there is no joint at C, will, 
oppose its resistance to bending to the force 5 ; but there will 
be a tendency to rise at the apex/?, if the fastening be not made 
sufiBciently strong. 

2P 




Fig. 125. 

IV. Third Roof with Single Trussed Principals. Fig. 124. 
In this form of truss, frequently known as the Belgian or 
French truss, the single vertical rod of the preceding form is 
replaced by a triangle BCD. The struts are placed in the 
middle of the rafters and the external forces are distributed as 
shown in the figure. In the force plan « 5 c = 3 /", and i, and 
2 are determined as before. By the decomposition of the re- 
sultant of I and 2 P, we obtain the forces 3 and 4, or d e and 
b e, and from the resultant e c, of the forces 2 and 3, we get the 
tensions 5 and 6, in c f and e f. The second half of the dia- 
gram is the sj'mmetrical counterpart of the first. 

V. Roof Truss with Double Trussed Principals. Fig. 125. 
This construction does not differ greatly from that shown in 
Fig. 124, except that the struts employed to strengthen the 



THE CONSTRUCTOR. 



17 



■rafters are divided into two. The spaces are equal to each 
other and the load uniformly distributed. As shown in the 
figure this gives a reaction of 5 P, or A and D. In the force 
plan a d = ^ P, and lines parallel to A E and A C drawn, de- 
termining the forces i and 2, or d e and e a. We then combine 
£ a with a ,b = 2 P, and decompose the dotted resultant e b, 
into the thrusts ^yaud f b, or 3 and 4, by drawing these lines 
parallel to jE Cand £ F. Again we take the resultant of the 
forces 4 and 2 P, and decompose it into 5 and 6, or f g and 
^ c, which brings us to the middle of the symmetrical figure. 
The force 7 is the resultant of 6, and its counterpart S, and the 
load 2 P, and the half of this force is therefore equal to the pro- 
jection of 6 upon the vertical, less P, or in the diagram, to d li. 





Fig. 126. 

VI. IJnglish Roof Truss, with Multiple Trussed Principals, 
yig. 126. Here we have inclined struts, with vertical tie rods. 
The load is again uniformly distributed, each space bearing the 
load of 2 P. The reactions at A and D are each = 7 /*. In 
Ihe force plan we have ab-\-bc-\-cd-\-de = z X 2 P -\- P=^ 
7 P, which gives the length of a e. The forces i and 2 are 
found by drawing/ « and e f, parallel to A E and A L. Now 
consider I as combined witli a b ^=1 P, and the resultant /' b, 
■decomposed into f g and g b, giving the forces 3 and 4 ; again, 
•combine 2 and 3, and then decompose the resultant g e, into 5 
and 6, or g h and h e, by drawing these latter parallel to L F 
and L M. In this manner we continue until we reach 12, or 
I d, which we then project upon the vertical. Now taking from 
d m, one-half the load P= d e, we have m e for one-half the 
stress on the middle rod B C. The remaining half of the force 
plan is similar. 



D___K^^^ 




Fig. 127. 



VII. Polygonal or Sickel Shaped Roof Truss. Fig. 127. 
This roof may be considered as a modification of the preceding 
form, and is used for higher and wider spans. It is hardly 
proper to assume that the load is here uniformly distributed 
even if the spaces are equal, for in the case of snow, much less 
weight would be carried by the steep portions A B or G H, 
than by the flatter surfaces C D or D E. We must therefore 
estimate the forces P-^, /\, P-^, acting as B, C, D, E, F, G, and 
make the reactions at A and B equal to Q ^ P^ -{- P., -\- P-^. 

In the force plan a b ^ P^, b c = P,. c d =P,,, and a d ^ Q, 
"which is first decomposed into i and 2, by drawing e a and d e 
parallel to A B and A J. Then combining i with /\, and de- 
composing the resultant, as before, we get 3 and 4, or ^ y and 
f b. Having 2 and 3, we get in like manner 5 and 5, ox g_f 
and d g ; then combining 4 and 5 with P^, and decomposing 
with parallels to C A' and C/?, we obtain the forces S and 9, 
and so proceed until we reach 12, which is the middle of the 
symmetrical figure. The members K L, D L, E L, and 3'T L 
are all subject to tension. 

? 51- 

The Graphicai< Determination of Wind Stresses. 

In designing large and important roof trusses it is important 
to investigate the stresses due to wind pressure, as well as those 



due to the weight of the roof and of snow, and indeed, in some 
cases, the resistance to wind is the most important of all. 

As an illustration of the applicability of the graphical method 
to the determination of wind stresses, we will take the English 
roof truss. Fig. 126, whose conditions under a vertical load have 
already been examined, and consider it as also subjected to a 
wind stress JV, as shown in Fig. 128. 

We have first to determine the forces Q-^ and Q^, acting at the 
points A and D. The wind pressure will be taken as acting on 
the surface of the roof from A to B. l,et IV be the resultant of 
the entire wind pressure acting normal to AB, and let Phe the 
total vertical load upon that half of the truss. By combining 
these two forces we obtain the direction FS of their resultant, 
and also its magnitude, which we then lay off on the force plan 
at cc-i- Upon the other half of the truss we have only the verti- 
cal load, which may be considered as acting aty, and equal in 
magnitude to P. By prolonging its direction until it intersects 
the previously determined line at S, we have at 5 a point in the 
resultant of the entire load upon the roof, including wind pres- 
sure. By making c^ a^ in the force plan equal to P, we have a c 
for the direction of this resultant, which may then be laid off at 
5 T'in the drawing. In order to determine the forces Q^ and 0.^ 
we must recollect that, according to \ 34, when we have two 




Fig. 128. 

closing forces to determine, we must also have at least two con- 
ditions given. In this case, then, we must first find the direc- 
tion of 0■^ and Q^. 

The wind pressure produces a horizontal thrust which must 
be met by the stability of the walls or columns upon which the 
roof rests. In each case it must be determined whether this 
horizontal thrust is borue equally or unequally by both sup- 
ports, and in what proportion it is divided. To this end we 
first find (according to \ 39) the proportion of the vertical com- 
ponent of the force a c, which comes upon each support (as 
found by the intersection of 5" T, prolonged with A £>), and 
then combine these vertical forces with their respective hori- 
zontal components. It often happens that all the horizontal 
thrust is borne by one of the supports, which it must of course 
be prepared to resist. This often occurs in the case of railway 
stations, and under such circumstances the direction of each force 
must be determined separately. First prolong the vertical at 
D downward until it intersects .S T, and join the intersection 
with A (the lines are only indicated in the figure). This gives 
the direction of the force at A. We have now both the direc- 
tion of the reaction at D and the direction of that at A. We 
must also consider the distribution of the forces at the various 
knots between A and B, and between B and D. We have for 
the points between A and B the resultants between the propor- 
tional parts of P and JV, while from B to D we have simply the 
proportional parts of P. This gives at A the force P^ ; aX E,F 
and G, the force P„ ; at the peak, the force P-^ ; aX H,J and K, 

P ' P 

the force />,= — , and at D, the vertical force P^ =--. 
4 f' 

Returning now to the force plan, we make c d =^ P^, d e ■= 
e f=fg = P,, gh= P.Jii = ik = k 1= P^, and la = P^. 
We now have finally the length b 1, for the value of the reaction 
Q.^ at the point D, and a line (not shown) from b to d, gives the 
magnitude of the force Q-^ acting at A. 

The determination of the stresses in the various members can 
now readily be made. The decomposition oi bd hy drawing 
bin and 7« </ parallel respectively to A E and A L, gives the 
forces I and 2. We thus proceed until we reach the rod B C, or 



38 



THE CONSTRUCTOR. 



No. 13, for which we get the tension r s =^ 13, by drawing the 
vertical r s from ;•, until it intersects the line n 5, drawn parallel 
to B D. We then continue to determine the forces from 15 to 
25, as already shown. The force plan shows that under these 
conditions similarly placed struts are subjected to dissimilar 
stresses. The determination of the stresses might have been 
made in the reverse order, beginning with the triangle x b /, 
•which should give the same results, and which may be used to 
prove the accuracy of the work. A proof is also made by the 
accuracy with which the line w x drawn from w, parallel to 
K O, intersects the point x, which was first determined by the 
intersection of b x and / x. As a matter of fact, it will be found 
to require careful drawing in order to insure the closing of the 
diagram. 

By comparing the last force plan with that found for the same 
roof truss in Fig. 126 ^the scale being the same), it will be seen 
how greatly the wind stresses affect the structure. In order to 
complete the calculatioa, a second plan should be drawn, as- 
suming the wind to act also upon B D. 

I 52- 

Force Plans for Fr.^med Beams. 

Beams of various forms are often framed in various shapes 
and made both of wrought and cast iron, and have many appli- 
cations, such as walking beams for steam engines, for cranes, 
arms, &c. A few examples will show the method of investiga- 
tion for such cases. 




~«=^ 



Fig. 129. 



I. Projecting Frames with straight members. Fig. 130. The 
load /'acts at ^ in a direction normal to the axis of the frame, 
which IS supported at B and C. The force plan is constructed 
as follows : Draw a b = P, and from its extremities draw a c 
and b c parallel to i and 2, which gives the forces in those mem- 
bers. Each of these is then decomposed into two other forces — 
I into 3 and 4, 2 into 5 and 6, giving the triangles bee and ad c. 

The forces 3 and 5 are then combined and the resultant de- 
composed into 7 and S. To do this we transfer $■:= d c iofe, 
and join the resultant y' 5, which can readih' be separated into 
7 and S. We proceed in this manner for the remaining mem- 
bers, and as the frame is symmetrical about the axis g c, only 




Fig. 130. 

one-half of the diagram need be completed. The lines ^ a and 
b g, which are the final resultants of 15 with 17, and 16 with iS, 
are also the external forces at B and C, the points of attach- 
ment, provided that their direction be permitted to remain the 
same. 

II. Double Loaded Frame, Fig. 132. In this case we have 
the force Pi acting downwards at ^ , and a force P„ acting up- 
wards at D, while the points of attachment remain at B and C 
as before. The members A B and A Care polygonal formed. 
The force plan is drawn just as before, until the force 13 is 
reached. At D the members are attached to each other at their 
intersection, so that the force P.^ acts upon both 15 and 16. At 
this same point we have the action of the forces 12 and 13. 
Now join the extremities of 12 and 13 by the dotted line shown. 



and mark off the length of the force P^, which is subtracted, 
because its action is upward, thus obtaining the resultant of the 
three forces. We can then draw 15 and 16 and proceed without 
interruption to 20. Finall3-, we draw b/and e a, the external 
forces at O^ and O^, which hold the entire frame in equilibrium, 
in. Framed Boom for a Crane, Fig. 133. This figure is por- 
tion of a framed arch which may be used for the projecting 
boom of a large crane. At A and D we have the forces F-^ and 
P.,, and at B and C, the external forces Q^ and O.^. The force 
plan is now required to determine the internal forces acting on 
the various members of the structure. Before this can be done. 




Fig. 131. 

we must first determine the as yet unknown direction of the 
force O-i- Prolong P^ and P, to their intersection at E, and by 
drawing in the force plan, the triangle a b e, determine the di- 
rection F E of their resultant ; then prolong O-^ until it inter- 
sects/? 7^ at G, and join C G, which will be the required di- 
rection of the force Q.,. Completing the figure in the force 
plan, we have c d = Q^ and d a-= O^. We now proceed from 
Pi=: a b and lay off the forces i and 2, decompose 2 into 3 
and 4 ; combine 3 and i and decompose their resultant, obtain- 
ing 5 and 6. We thus proceed until we reach 12, which we 
obtain by combining 9 and S and decomposing the resultant 
into II and 12. We now have to combine 10 and 11 with P^, 
and decompose the resultant into 13 and 14. We first transfer 
the force 11 to 'e, making it equal to ef, in order to avoid the 
confusion of lines, which would occur if the construction were 
made at a. Now drawing the path 11, 10, P^, we have the clos-. 
ing line cf, which decomposes into 13 and 14. We then have 
15 and 16 from the resultant of 13 and 12, and finally, 17, as the 
line joining 15 and 16 with rf, since 16 and 17 must have the 
resultant a d = O,- If the work is correctly done, we will find 
17 falls parallel to B C, which affords a convenient and valuable 
proof for the whole work 

?53- 
Rem.\rks. 

The foregoing problems and methods will serve as general 
examples of the various applications of Arithmography and 
Graphostatics, and at the same time it must be noted that great 
care and neatness are most essential in the use of the method. 

It may be added that it is desirable to use as few letters and 
figures as possible in designating the various lines ; a common 
fault of beginners being the disfigurement of their work in this 
respect. The necessary marks should be made quite small and 
in faint pencil, so that they may be readily erased if so desired. 

It is necessary also to be provided with the best grades of 
pencils, well sharpened, a good drawing-board, reliable pro- 
tractor scale, dividers, and flexible spline ; and it is the author's 
experience that these cannot be used too carefully. In order to 
acquire facility in the methods and confidence in the results, 
the beginner is advised to begin with simple examples which, 
can be thoroughly understood, and practice upon these care- 
fully. 

By proceeding in this manner it will be possible to obtain a 
skill and grasp of the graphical method which will enable the 
student to use it freely for the solution of a great variety of 
problems, and extend its scope far beyond the range of the ex- 
amples which have been given. 



THE CONSTRUCTOR. 



29 



SECTION III. 



THE CONSTRUCTION OF MACHINE ELEMENTS. 



Introductory. 

Under the title of "Machine Elements" we may consider 
those single or grouped parts which are employed to a greater 
or less extent in all forms of machinery. It is not practicable 
to determine their number, nor, indeed, is that a matter of im- 
portance, since the selection of groups and details is not based 
upon any positive or generally accepted system. The following 
selection of the constructive elements of machinery may be 
found useful and convenient, which is the principal end to be 
attained. 

la the previous sections a number of general formula have 
been given, while in the cases which follow detailed examples 
are selected. The dimensions and weights are expressed in 
inches and pounds, except where otherwise distinctly stated ; 
velocities in feet per second ; and rotations in turns per minute. 
The measure of force is the pound ; that of work in foot pounds 
per minute, or for larger quantities in horse-power — (33,000 foot 
pounds).* 

CHAPTER I. 
X/yET/JVC. 

\ 54- 
Rivets. 

Rivets are principally used for joining sheet metals or other 
flat shapes together for the construction of a variety of sheet 
and framed structures. They may be considered as a funda- 
mental machine element acting to transform detailed parts into 
combinations. 

In the illustrations various forms of rivets are shown. The 
common wrought-iron rivet is shown in Fig. 132, with the 




Fig. 132. Fig. 133. Fig. 134. Fig. 135. Fig. 136. 

button head, while Fig. 133 shows the conical head generally 
formed by hand riveting. The length of body required to form 
the head varies from 1.3 to 1.7 times the diameter, according to 
the completeness with which the rivet fills the hole. When the 
head is formed by dies instead of the hand hammer, the shape 
is usually conoidal or spherical, as in Figs. 134-135. 

The slight bevel given to each end of the rivet, as shown in 
Fig. 13S, adds materially to its strength. The double conical 
hole shown in Fig. 137 assists in uniting the plates, and this 
shape may be produced by using in the punching machine a die 
slightly larger than the diameter of the punch. This difference 
tias been experimentally determined for wrought-iron plates, 
and is secured by making the hole in the die equal to the diam- 
eter of the punch plus '4 the thickness of the plate. 

In Fig. 136 is shown a form of countersunk rivet used in 
shipbuilding. 

For bridge construction great care should be taken in the 
choice of proportions. Figs. 137-139 show the proportions 
adopted for the Dirschauer Bridge after the careful researches 
of the engineer Kriiger. Fig. 137 shows the normal rivet head, 
and Figs. 138 and 139 the half and full countersunk heads. 

Rivets up to I or I ^ inches in diameter may readily be closed 



* These quantities are all given in metrical units in the original, but have 
been transformed in the text into Eng^lish t'uits. It must be remembered 
that the metrical horse-power (75 kgm.) is slightly smaller than the English 
horse-power. — Trans. 



with hammers of S to 10 pounds weight ; but if the head is to 
be formed in a swage or die, a heavier hammer, say 16 pounds 
weight, is necessary. 

The rate at which this work can be done by skilful riveters 
per day, according to Molinos and Pronnier, is as follows : 

Diameter of Rivet. No. per Day 

yi" 200 to 250 

^yl" . . 180 " 200 

fs," . 100 " 125 

\" 90 " 100 

These figures are for horizontal bridge work ; on vertical 
members about three-fourths these numbers may be taken. 




Fig. 137. 



Fig. 13S. 



Fig. 139. 



Much higher rates are shown upon boiler riveting, as may be 
seen from the following table, based upon observation of eleven 
days' work at the boiler works at Piedboeuf (Aachen) : 



Diameter of Rivet. 



^•8 

H 



I/S 

lA 



No. per Day. 

■ • 350 

• • 325 

• • 300 
. . 2S0 
. . 260 
. . 240 
. . 220 
. . 200 



In cylindrical shells of more than three feet diameter these 
rates may be increased ten per cent., while for awkward or diffi- 
cult work ten per cent, reduction should be made. Each man 
had the assistance of tv?o strikers, one holder and one boy, 
sizes less than y%" requiring but one striker. 

Hand riveting is now being largely superseded by machine 
work. These machines possess the advantage of performing 
the work much more rapidly, thus insuring a stronger joint, be- 
sides which they are much more economical. Since their first 
introduction at the time of the building of the Conway Bridge, 
they have been extensively used for bridge work, and with the 
improvements which have been successively made they are 
rapidly displacing hand riveting for boiler work.* 



^^ .^mong modern riveting machines the hydraulic riveter of Tweddell 
holds the "first place. For a description the following references may serve : 

Polyt. Zentral bl. 1874, p. 103 ; Engineering, Jan., 1875, p. 76 ; Sellers' Im- 
proved Tweddell's Machine, J^ur. Frank. Inst., 1S76, p. 305 ; Tweddell's 
Machine on a Crane, Sci. American. 1S76, p. 226 ; Small Tweddell Machine, 
Revue ludust.. 1S76, p. 349. 

Other forms of steam and hydraulic riveters well suited for boiler work 
are ; 

Garforth's Machine, Kronauer's Zeich. III., Johnson's Imp. Cyc, PI. 42 ; 
the excellent steam riveter of Gouin, see Molinos & Pronnier, Ponts Metal- 
liques, p. iSo ; also the hydraulic riveter at Creusot (giving a pressure of 20 
to 80 tons on the rivet, and closing 2 to 25 heads per minute). Revue Indiist., 
1S75, p. 349 ; also the very heavy machine of Kay & George (giving a pres- 
sure of 120 tons on the rivet), Engineering, 1S75, p. 223. .A.n apparently very 
ingenious machine is that of -\llen, used especially for boiler riveting. In 
this machine the frame and post are temporarily held together by a rod 
operating in a very ingenious manner through one of 'he open rivet holes. 
At the Philadelpliia Centennial Exhibition this machine closed three or 
more rivets per minute. 



40 



THE CONSTRUCTOR. 



«55- 
Strength of Riveted Joints. 

Riveted joints are intended either to resist direct stresses (as 
in bridges and similar structures), or to secure a tight joint 
against moderate internal pressure (as in ships, gasholders, 
etc.), or in most cases both these conditions are united (as in 
the case of steam boilers). A distinction may then be made 
between joints for strength and joints for tightness, the seams 
of steam boilers standing midway between the two. 




Pig. 141. 



Fig. 142. 



Joints for strength are either simple lap joints, Fig. 140, or 
butt joints, Fig. 142, the latter coming into general use for 
bridge work. The joint shown in Fig. 141, called a flap joint, 
is also somewhat used for vertical tubes, chimney stacks, etc., 
the flap really being nothing but a narrow plate. 

For any given thickness, (!, of plate it is impracticable to 
make the riveted joint the same strength as the plate itself, but 
the ratio between the strength of the plate and the strength of 
the joint can be made a maximum. This will best be attained, 
witii the assumption of a sufiicient margin, when the strength 
of the rivets and the strength of the remainder of the metal 
between the rivet holes are eqtial to each other, i. e. , when they 
reach their limit of elasticity at the same time. If the rivets 
and plate are of the same material, we have, according to \ 5, 
the stress in the cross section of the rivets as 0.8 that of the 
plate. From this we derive the following formula, in which 
the friction of the joint is neglected, as being of uncertain 
value : 
Let- 
ts = the thickness of the plate, 
d -^ the diameter of rivet, 
a = the pitch of rivets, i. e., the distance from centre to 

centre of adjacent rivets, 
n = the number of rows of rivets, 

(p = the modulus of resistance of the joint, being the ratio 
of the resistance of the joint to that of the full plate ; 
then the highest ratio of resistance will be attained when we 
have for lap joint riveting : 

1 



(43) 



a 


5 \ ^ J 


+ 4 


which gives : 


d 






a 

1 + 


I 5 'S 

n TV d 


or for butt joint riveting: 




a 
~6~~ 


■K f d 

5 V S 


)V4 


•which gives : 


d 


I 




a 

I + 


•2. 11 7T d 



(44) 



For lap joint riveting : 

b' 5 a — d 7 

-^- = ^0-5-1- 0-56 y - 

For butt joint riveting : 
b' 5 « — d TT 



(45) 



;i 6 



= ( 0.5 + 0.79 



/4 



In both cases a good value of b, in practice, giving sufiScient 
room for rivet heads, will be secured by making : 



6 = 1.5 d, or -,- = 1.5 -r 



(47) 



A point of interest is the superficial pressure p, which exists 
between the body of the rivet and the cylindrical surface of the 
rivet hole. If S^ is the stress in the punched plate we have — 

For lap riveted joints : 

(48) 



Jl — ^ 

S2 ' " (! 

For butt riveted joints : 
p d 



(49) 



The following table and scale will serve to reduce the numer- 
ical labor of these calculations : 



56. 



Table and Proportional Scale- 





1.0 


1.5 


2.0 


2-5 


3-° 


4.0 


71 = 


z 


2 


I 


2 


1 


2 


I 


2 


I 


2 


1 


^2 


a 

S 

. // 

.s « ~ 

£, i" 
^ * = 

5. 


1.63 

0.39 

1.06 
0.39 
0.63 


2.22 

0-39 
1.06 

0-5S 

0.63 


2.92 

0.88 

1.78 
0,49 
0.94 


4-33 
0.88 

1.78 
0.65 
0.94 


4.52 

1-57 

2.5S 
0.56 
1.26 


7.04 
1-57 

2.58 
0.72 
1.26 


6.43 

=■54 

3-46 
0.61 

1-57 


10.37 

2-54 

3-46 
0.76 

I-S7 


8.67 

3-53 

4-31 
0.65 
1.88 


14-33 

3-53 

4-31 
0.79 

J. 88 


14.07 

6.28 

6.48 
0.72 
=-51 


24.14 

6.28 

6.43 
0.83 
2-51 


a 

^ b' 

S i, ~ 

A i" 

« * = 


2.25 

0.79 

1.29 
0.56 
1.26 


3-5= 

0.79 

1.29 
0.72 
1.26 


4-33 
0.96 

2.20 
0.6s 
1. 88 


7-15 

0.96 

2.20 
0.79 
1.88 


7.04 
3-14 

3.24 
0.72 
2. SI 


12.05 

3-14 

3-24 
0.83 
2.51 


10.37 

4.91 

4-37 
0.76 
3.14 


18.21 

4.91 

4-37 
0.86 

314 


14-33 

7.07 

5.60 
0.79 
3-77 


25.61 

7.07 

5.60 
0.90 
3-77 


24-14 

12.56 

8.32 
0.83 
5.03 


44.21 

12.56 

8.32 
0.9.;. 
S-Q3 



The overlap of the plate is subjected both to shearing and 
bending. For the former conditions, call the lap b' , and for the 
latter h" , measuring in both cases from the centre of the rivets 
to the edge of the joint. To obtain the same resistance in the 
lap as in the perforated portion of the plate we have — 



(46) 



In the proportional' scale, Fig. 143, the principal values are 
graphically shown. It will be seen that the higher ratios of 
strength are not very practically obtained, for the large diameter 
rivets are inconvenient to handle. The advantages of lap joint 
riveting are also shown. The objection to butt joint riveting, 
which overrules its advantages, lies in the rapid increase in the 

d 
value of/, as it will be seen that with a ratio-T-=3 the elas- 
tic limit of \','rought iron is exceeded, the stress reaching 30,000 
lbs., while the stress upon the metal between the holes is only 
8,600 lbs. This explains the failure of riveted joints under 

d- 
variable tension loads. If the ratio of — j- = 2 is used, the 

excessive stress in the rivet holes cannot occur. Fairbairn, 
upon whose experimental researches these conclusions are 
based, states that for riveted structures the diameter of rivet 

may best be taken as equal to —7" X ^ ; but this conclusion is 

not fully borne out by experience. The use of the value for the 
lap (5 ^ 1.5 d is approximately correct at least for lap joint 
riveting, as the diagram shows it to give a slight margin both 
over the values of 0, for shearing or for bending. 

?57- 
Riveting Disposed in Groups. 

If more than two rows of rivets are to be used the efSciency 
of the joint may be decidedl}' increased without using incon- 
veniently large rivets by disposing the rivets in groups on 
either side of a central row, arranging them according to an 
arithmetical series. 

The numbers in adjacent rows may then be placed as follows ; 



: 2 : I 
13:2 

: 4 : 3 



Total 4 

" 9 
" 16 



THE CONSTRUCTOR. 



41 




Fig. 143. 

The following illustrations show examples of this form of 
riveting, which may be termed Group Riveting, and is espec- 
ially adapted to lap joints. The dotted lines show the limit of 
the area including each group, and the spaces between the 



a TV 1 d \- \ 


6 


or 




a TV f d \ . I 
d 5 V tJ / "' 





If now we assume that the force P, upon each 
o.Sit strip between the dotted lines, is equally divided 
among the rivets, we have for the efficiency of 
"■8^ the first row : 



(50) 



If the stress in the punched plate in the lines 
I, II, III, IV, etc., Fig. 146 be called S\, S^l 
5',", 57, etc., we have: 

P=S\ [ina — d)i 

= S-l{.na-.d)-^-^ 

-'■'^' ima — Hid 

111 



= 51 



m- ■ — 6 



■5d) 



in' — 10 



•(51) 



And from this when S\ = S^^ we have : 

a VI- + 1 

(/ in 

And upon the same supposition : 



(52) 



5 



IV 



m'' — 6 



m' — 2 

c-V 



W — 3 



St 



4' 



MT HI iv: V 






o|o 




o-<?jo 

O ■<?f o 
O ..^.tO 

""o'S'o"' 
060 
_o 6 o __ 

060 
o 6 o 



6 <?■??? 

O Too 
j -0' 



9 6 9 
It 



o 



o 



(5 . 
Hi? 



o 



VI VII. 



Q 



<P 



that is, the stresses at the lines III, IV, V, are 
smaller than 5^ = 6'". The useful application 
of this fact may be readily seen. 
Let us introduce -n (50) : 

-^ = ^= 1.5916, or say 1.6 . .(53) 

that is, we make the ratio d : (S constant and = 
1.6. For the modulus of the efSciency of the 
joint <P, when the stress in the solid plate is S^, 
we have : 

^ = A = i_^=-^. . (54) 

We also have for the pressure /, on the rivets : 
p^-^,= S\ (55) 

Tabtaating the results of the applications of these equations 
to various groups we have : 

j« = 2 3 4 S 



d 
& 


= 


1.6 


1.6 


1.6 


1.6 


a 
'd 


= 


2.50 


'h'hli 


4-25 


5.20 


a 

T 


= 


4.00 


5-32 


6. So 


S.32 



^ = o.So 0.90 0.94 0.95 

For joining narrow plates the rivets may often be disposed in 



Fig. 144. 



Fig. 145. 



Fig. 146. 



rivets in each longitudinal row are uniform. If in be the num- 
ber of rivets in the m'ddle row of each group, then the total 
jiumber of rivets in each group is ;«'.* 



ESEH 




5 -Oj O I 



010 -Oj 
1 Oo o 



Fig. 147. 



Fig. 14S. 



' This follows because, -^ Fi + m + -^ i + (,« — i) I ('« — ') = "'-• 



double groups, while for the union of several plates, as in the 
construction of plate girders, a number of groups may be em- 



THE CONSTRUCTOR. 



ployed. In Fig. 147 a triple row is shown, and in Fig. 14S a 
quadruple row, the joint in each case being made with a flap. 

Besides the advantage which results from the disposition of 
the rivets in groups in such cases, there is also a gain in making 
the flap somewhat thicker than the plates to be joined. The 
calculation in this case may be derived from the preceding 
methods ; also see the latter part of \ 59. 

Steam Boiler Riveting. 
For the joints of steam boilers parallel riveting is generally 
used. In this case the question of the tightness of the joint 



The following table contains the collected results of the pre- 
ceding formulse for the more commonly occurring proportions : 

?59- 

Tabi,e -iND Proportionai, Scai,e for Steam Boher 

Riveting. 




Fig. 149. 



Fig. 150. 



Fig. 151. 



prevents a wide spacing of the rivets. For the same reason the 
thinner plates require proportionally larger rivets than the 
heavier plates, and the lap of the plates, and also of the rivet 
heads, must be greater. The method of caulking is also to be 
considered. The older method consisted in driving the caulk- 
ing chisel into the perpendicular edge of the plate and forcing 
the lower edge of the groove thus made down upon the lower 
plate, as shown in Fig. 149. The modern method, shown in 
Fig. 150, requires the plate to be planed on the edge to an angle, 
which can then be caulked without grooving. The 
angle of bevel should be about 18^2°, or about i in 3. 
The method of the American, Connery, consists in the 
use of a round nosed caulking tool. Fig. 151, which is 
much less liable to injure the lower plate than the 
sharp chisels of the previous methods ; the action ex- 
tends farther into the plate also. 

The consideration of the tightness of the joint com- 
pels a modification of the theoretical treatment of the 
proportions of boiler seams, based upon practical ex- 
perience. 

According to Lemaitre the following proportions 
are suitable for single riveted joints : 







Round 
Head. 


Conical 
Head. 




c 


.5 


Modulus of Efficiency. 


s-s' 




d 








c 

,2 


JL) 

c 

'tSi 


«5 

1 
& 






It 


s 








5 


*' 


*" 


fc" 


^ 


Vi 


X 


n 


t\ 


X 


X 


iX 


iX 


0.66 


0.51 


0.60 


2 


A 


A 


X 


X 


y. 


X 


I 


iX 


2>^ 


0.66 


0.51 


0.60 


4X 


X 


A 


A 


1. 00 


A 


iX 


iX 


iX 


2X 


0.63 


0.48 


0-59 


6X 


t\ 


H 


n 


iX 


X 


iX 


>x 


iX 


2^ 


0.61 


0.48 


0-59 


13X 


y% 


W 


A 


iX 


A 


1% 


2 


iX 


2% 


0.60 


0.47 


0-59 


20 


A 


If 


r^ 


iX 


% 


iX 


2X 


2 


3X 


0.60 


0.47 


0-59 


29 


% 


if 


A 


-S-% 


X 


iX 


2X 


2X 


3X 


0-59 


0.47 


0-59 


36X 


A 


X 


n 


T% 


if 


2.00 


2X 


2X 


aX 


0.5S 


0.47 


0.59 


50X 


Y^ 


lA 


w 


2.00 


X 


2X 


3 


2X 


4X 


0.5S 


0.47 


0-59 


66 



d= i.5<T-f 0.16" 
a =: 2fi' -f- 0.4'' 
b = i.srf 



(56) 



Double riveted joints are also much used for steam 
boilers, especially for the longitudinal seams, while 
single riveting is used for the circumferential joints, 
since the stress upon the longitudinal joints is much 
greater than upon the circumferential joints. 

For double riveted joints, that is, for riveting in two 
parallel rows, we have for the pitch a.^ of the rivets in 
each row : 



a„ = yi -]- 0.78' 



(57) 



■while the space between the two rows may be taken 
as equal to the previous value of a, or 2d + 0.4''. 
In some cases this value is used for the pitch of both 
rows (see Fig. 153). 

We have previously taken the modulus of efliciency 
<t>, so that the rivets and the perforated plate have not 
the same degree of security. The values of 0, for the 
rivets and for the plate should therefore be determined 
separately, and the smaller value taken for that of the 
completed joint. Let : 

'P' = the modulus for the perforated plate, 
<p" = that of the rivets, 

then according to the previous formula; we have : 

a — (/ 



{58) 



(jl" z= 



p 



" liS J 

jpon th 
ingle an 



For the pressure p upon the body of the rivet we «S 
have finally, both for single and double joints. 



(59) 



The stress S^ in the perforated plates of boiler shells ^ 
is not generally permitted to exceed 5000 to 6000 
pomids per square inch of cross section. 




Fig. 152 



THE CONSTRUCTOR. 



43 



The rivet length is = 2;? + i.yrf, upon the assumption that 
both plates are of the thickness i5, and this length gives an 
ample allowance for the full clearance of the rivet holes (see 
§ 54). The last column is of service in making estimates of 
weights. 

Fig. 152 is a graphical presentation of the principal results 
of the preceding formula;. It will be noticed that for single 
riveting the modulus 1^", for the rivets, is always less than the 
modulus !]>' for the perforated plate, and is nearly always less 
than j^. It follows that for single riveted joints of steam 
boilers we should never assume a greater strength than one-half 
that of the solid plate. By the adoption of double riveting, 
while retaining the same pitch, rz = arf + 0.4'', we ought to 
obtain, according to the formula of \ 5S, a value of <\i" , twice 
as great, which in the case of very light plates would exceed 
unity. In that case, however, the value of 1^' is the lesser, and 
it determines the efficiency of the joint, so that the only gain 
due to the double riveting in that case is the increase in the 
value of <?'. If, however, we choose for double riveting the 
pitch a.,, as given from equation {57), both 1^' and <)" will be in- 
creased in value. The lesser of the two moduli is that for the 
rivets, and its value is obtained from 



T d- 
5 ai 



(60) 



Its value lies between 0.75 and 0.59, and is shown in the table 
and diagram. The pressure/ in all cases remains within prac- 
tical limits. 









! 
1 


-©-,- 


^ i ^ = 







ll-c^-r 


1 


i ii. 


1 © \ 






i :: 


Q 

[ 
t 
i 
i 



Fig. 153 



Fig. 154. 



Fig. 155. 



In Fig. 153 is shown double riveting in which the pitch kept 
equal to "id -f- 0.4", while in Fig. 154 the pitch is made equal to 
j,d 4- 0.78" for rivets in the same row, while the diagonal dis- 
tance between rivets of the two rows is the same as for single 
riveting. For a flap joint such as is shown in Fig. 155, we have 
a combination of parallel and group riveting. This method is 
used in Germany for steam boilers, but is little used in America, 
if at all. The flap is placed on the inside of the boiler shell, 
and the flap seams have only half as many rivets as the main 
seam, but of the same diameter. The objection that the inner 
edge of the main seam is made inaccessible is counterbalanced 
by the increase in strength. AVe have 

ia — d 
y = .... 



y = 



0.3 IT d'' 



ad 



the lesser of which -will be found to exceed the value obtained 
for ordinary double riveted joints.* 



Example. S = -^^", d = ^i", a 
- „ Q, A" _ °-3'^X 0.391 



- 2d -\- 0.4" = 1.65'', say i;^'s" ; </>' = 



3-3 



„.. f - , , —0.71. It may be remarked that American prac- 

1.65X0 3125 
tice gives wider pitches than are generally used in Europe. f 



Three rows of rivets are used in this form of joint, and the 
outside rows of wide pitch make this method more trouble- 
some of execution than the group riveting shown in Fig. 144, 
which has a modulus of o.So. This is a point which should be 
borne in mind. 

The joints of gasometers exhibit but little variety in plates or 
rivets. The rivets are usuallj- about '4 " to yV" in diameter and 
i" pitch, with a lap at the joint of about j4^', the rivets being 
closed cold and the joints caulked with red lead. 



Table of the Weight of Sheet Metal. 



Thickness 

in 

Inches. 



Weight in Pounds per Square Foot. 



I 
Wro't Iron! Cast Iron, 



Coppc 



Zinc. 



tV 


2-53 


2-34 


'A 


5-05 


4.69 


z\ 


7.5s 


7-03 


'/ 


10.10 


9-3S 


A 


12.63 


11.72 


Vi 


15.16 


14.06 


A 


17.68 


16.41 


'A 


20.21 


1S.75 


A 


22.73 


21.09 


A 


25-27 


23-44 


H 


27.79 


25-7S 


H 


30-31 


28. 13 


il 


32-S4 


30-47 


A 


35-37 


32.S1 


if 


37-90 


35-16 


I 


40.42 


37-50 



2-73 


2.89 


3-71 


5-47 


5-73 


7.42 


8. 20 


S.67 


11.13 


10.94 


11.56 


14.83 


13-67 


14-45 


18.54 


16.41 


17-34 


22.25 


19.14 


20. 23 


25.96 


21.88 


23-13 


29.67 


24.61 


26.02 


33-38 


27-34 


28.91 


37-oS 


3.x oS 


31.80 


40.79 


32.81 


34.69 


44-50 


35-55 


37-58 


48. 21 


38. 28 


40.47 


51-92 


41.02 


43-36 


55-93 


43-75 


46.25 


59-33 



2-34 
4-69 
703 
9-3S 
11.72 
14.06 
16.41 

18.75 
21.09 

23-44 
25-78 
28.13 

30-47 
32-81 
35-16 
37-50 



I 61. 

Special Forms of Riveted Joints. 

Junction of Several Plates.— In Fig. 156 is shown the junction 
of three plates. In this case the corner of sheet No. 2 is bev- 
eled off and No. i worked down over the lap. 



ljfiSS^?l*'V-Ji !•««,* K.'A 


I 3 


i Q 
J Cp 


2 


■ef-e-^- 


-0--0- 


1 3 





i 

„.-' 


9 
i 



2 


s 


■-Q-i-QrX^- 


-.e--0-| 




3 if 



Fig. 156. 



Fig. 157. 



In Fig. 157 the junction of four plates is shown. Here the 
angles of sheets Nos. 2 and 3 are beveled and Nos. I and 4 are 
left unaltered. In the construction of steam boilers the shell 
may be formed either in cylindrical sections, as shown in Fig. 
158, or in sections of a conical shape, the taper of all the sec- 



* The two rivets -^vhich lie between any pair of rivets in the main joint 
each bear a stress of \ P, and the rivets of the main joint also sustain a P- 
The flap does not transmit an^- stress to the rivets of the main seam For 
the stress on rivets P^= ^ T^d- S'-^, for the solid plate P^ Si 2aS. The modu- 
lus 0", taking S^ for shearing stress = . }., , is found from <}>" = -^-^^-5 

— ' For the main seam we obtain tf)', from P= ^ (■2it — ^d) S S"^ = 



zad 
Si 2a8, whence q 



3 ('^ - '^) 



■which is greater than - 



t For example, in single riveting ^s" plate and fj" rivets have j%" pitch 
(the formula would give about i^4")> A" plate and Vs" livets have a pitch of 
zfs" (the formula would give about 21^5'') ; in double riveting, for %" plates 
with y^" rivets, the pitch would be ^yi" , while the formula would give 
about 3". 




Fig. 158. 



Fig. 159. 



tions bearing the same relation to the direction of the flame as 
shown in Fig. 159. This latter method requires that a slight 
curvature should be given to the sheets in order to secure the 
required taper. The determination of the taper and curvature 
of the sheets and lines for the rivet holes may be made in the 
following manner : 



44 



THE CONSTRUCTOR. 



Let— 

D = the diameter of the shell, as in Fig. 159, 
J) = the breadth of the sheet, Fig. 160, on a circumferen- 
tial seam, 
/, ^ the length of the sheet between pitch lines of rivets, 
/== the versed-sine of the arc £ ; we then have : 



/ 
6 



~D L 



(61) 




Example, 
we have B - 



Fig. 160. 



In a riveted tube where each section is made of an entire sheet 
= -nD. If the breadth B is twice the length /-, we have 



- = 0.7854 X 2 = 1.570S, 



so that f will be a little greater than i>^ times the thickness of the plate. 

In arranging the junction of sheets when the flap joint is em- 
ployed, care must be taken to avoid complicated intersections. 
This is best accomplished b}- making the flaps on the longitud- 
inal and circumferential seams come on opposite sides of the 
plates. Where the flaps are both on the same side, they are 
sometimes let into each other. 

Rciiiforcevient of Plates. — This may often be done verj' 
readily by the use of angle and T iron. In Fig. i6i is shown 



X.1* 



it'jL>=ll-r«,S tfx 

Fig. 161. 



-}. 



,1 



Fig. 162. 



-^1 = 13 T 8 rf — 

Fig. 163. 



J.I J 



an internal angle iron, and in Fig. 162 an external, and in Fig. 
163 a simple! iron. The proportions for angle iron given by 
Redtenbacher are as follows : 

h = height of angle arm, 
(i = thickness. 

h = 4.5 '' + I". 

For T iron h^ = the base := 8 d + 2", and the height of the 
rib = ){/ii. In practice a great variety of proportions are made 
to suit all possible cases, examples of which may be found in 
the illustrated catalogues of the mills where they are rolled. 



tected from corrosion by being incased in copper. Screw stay- 
■ bolts are now often made of soft wrought iron or mild steel, 
but copper bolts are still preferred by many. 





Fig. 167. 



Fig. 16S. 



Fig. 169. 



jdiG. 170. 



Construction of Angles (Figs. 167-170). — Angle junctions in 
riveted work are made either by flanging the plate or by the 
use of angle iron. In Fig. 167 the flange is turned inward, and 
in Fig. 1 58 it is turned outward. In these cases h is made the 
same as for angle iron of the same thickness. Figs. 169 and 
170 show the use of internal and external angle iron. 





Fig. 171. 



Consti-Hction of Solid Angles. — These are the most difficult 
forms of riveted work, and may be made in several manners, 
the most important being shown in the illustrations. In Fig. 
171 the vertical angle is made as in Fig. 167, and the horizontal 
angles as in Fig. 169, sheet No. 2 being beveled under the angle 
iron. In Fig. 172 all three angles are made as in' Fig. 169, the 



S^ 



U 





Fig. 173. 



Fig. 174. 



vertical angle iron being cut and bent over the horizontal, In 
Fig. 173 the angles are all made as in Fig. 169, but the angle 
irons are welded together at their junction. This makes an ex- 
cellent piece of work, but is difficult and expensive, and re- 
quires iirm support for the work, and is only applicable for 
important constructious. In Fig. 174 the vertical angle is made 
like Fig. 169, while the lower joint is made as in Fig. 170, mak- 
ing simple and substantial corner. 





Fig. 164. 



Fig. 165. 



Fig. 166. 



The strengthening of parallel plates which are near together 
is best done by the use of staybolts. In Figs. 164 and 165 is 
shown 3 copper staybolt after and before riveting, this form 
being used in locomotive fire boxes and marine boilers. The 
central hole affords a warning of the corrosion or weakening of 
the bolt by the escape of steam. It is best to remove the screw 
thread from the projecting portions before riveting over the 
heads. Fig. 166 shows a form of iron staybolt for the same 
purpose. The short piece of tube between the plates prevents 
them from being drawn out of shape while riveting, and the 
opening permits a free circulation of water. The bolt is pro- 



THE CONSTRUCTOR. 



45 



CHAPTER II. 
HOOPING. 

I 62. 

Hooping by Shrinkage. 

The use of hoops or bands is a very e£&cieiit method of unit- 
ing some combinations of machine elements, and also for 
strengthening existing combinations. The hoops or bands are 
arranged so as to encircle the portions to be united, and caused 
to exert sufficient pressure upon them to create such friction 
between the surfaces as to prevent any relative motion. It fol- 
lows that the material in the band is subjected to tension while 
the parts which are held together are under compression. The 
bodies to be hooped are nearly always either cylindrical or 
conical in shape. 

The pressure required to secure the hoops maj' be obtained 
either by shrinkage, a method formerly used very extensively, 
or by cold pressure, a modification being described in the latter 
part of I 64. 

The elongation which is produced by elevating the tempera- 
ture to a red heat may be taken for steel and wrought iron at 
about Tj^ij, while to keep within the limits of elasticity the re- 
sistance to contraction should be, for 

Cast or wrought iron TiVo 

Cast steel ^k 

Hence the allowance for shrinkage to be made in boring for a 
cast iron hub to fit over an unyielding centre should not be 



greater than 



and is best made from 



to 



T3B0 Lo 1 S50, especi- 
ally if the centre is very heavy. The ring can then be fitted to 
its place when at a dull red heat. For wrought iron or steel, 
rings, such as wheel centres, such precautions are not so essen- 
tial, since these materials permit of a slight extension without 



injury (see ■ 



If the centre possesses but very slight yield- 



ing elasticity, there may be danger, however, that the contrac- 
tion due to excessive cold may overstrain the material. 







Fig. 175. 

When wrought iron bands are to be used to secure iron jour- 
nals to wooden shafting, as shown in Fig. 175, the end of the 
shaft is made slightly conical, so that the bands, being raised 
only to a dull red heat, may be driven on with the hammer. 
The rings may be forged tapering, but the 'taper may be also 
readily produced by Clerk's method by repeated heating and 
cooling.* The red hot ring is immersed in the cooling tub for 



i<— d;-- -- ^-: 

kei i N'fii 


1 1 



h 

I 

Fig. 176. 

about half its axial height. The rapid contraction of one por- 
tion of the ring deflects the warmer portion towards the centre, 
and by repeating the process the taper may be produced to 
almost any extent which may be required. 

The following experiments, made in the Royal Technical 
Academy, will serve to illustrate the process. The ring shown 
in Fig. 176 had the following original dimensions : 

n = s'/y^, S = //', D = 8)4". 

After the first immersion the contraction was /^ 
" second 

" third 

" fourth 

" fifth 

" sixth 



5 
SB" 

7 
"J5 



1 6 
ST 



After the last immersion the dimensions were found to be /? == 
7jf at the upper edge, and at the lower edge = S/'j". 

A method of connecting two flat bars by shrinking on a hoop 
is shown in Fig. 177, and has been used at Seraing with good 
results. 

The hubs of gear wheels or revolving cylinders are advanta- 
geously strengthened by bands if they are cast in several parts, 
as in this way they are firmly united into a compact whole.* 




Fig. 177. 

§63. 
Cold Hooping. 

In the place of shrinking bands to their places, the more re- 
cent method of forcing them on cold has come into use for 
bauds of moderate size, such as for hubs of wheels, cranks and 
levers. In this case the ring and its seat are both made truly 
cylindrical, with merely a slight bevel for entrance, and then 
by means of a press forced together.f The difl'erence in diam- 
eter between the ring and hub is very small, and may be calcu- 
lated as described in ? 19. 

An investigation of the force required to push a ring on may 
be found desirable. The force which is necessary to press a 
cylindrical pin into a hub by continuous motion may be taken 
as nearly proportional to rate of progress, since it has to over- 
come the resistance of sliding friction between the surfaces. 
The pressure p, per unit of surface, is equal to the initial radial 
stress S], which exists upon the pin. If we make r the radius 
of the pin, / the length of the hole, y the co-efficient of friction, 
we have for a maximum value of the forcing pressure Q : 

Q= xrizlS^f {62)- 

Taking /= 0.2, as indicated by experience shown in the fol- 
lowing cases, we have 

5Q__ 

27rr/ 



P = S,: 



(63) 



For the tangential stress S.,, in the metal of the ring, we obtain 
from formula 37, I 19 : 






(64) 



And taking the thickness of the metal of the ring as (5, we get 
the value of p : 



I +■ 



i + - 



(65) 



+ I 



This gives for the following ratios of thickness to radius, cor- 
responding numerical values : 

— = 0.50 0.55 0.60 0.65 0.70 0.75 
p = 0.3S5 0.415 0.43S 0.463 0.4S6 0.508 

— = o.So 0.S5 0.90 1. 00 1. 10 1.20 

' i 

p = 0.528 0.54S 0.566 0.600 0.630 0.65S '■ 

— = 1.30 1.40 1.50 1,60 1.70 i.So 1.90 2.00 
r 

p = 0.682 0.704 0.724 0.744 0-759 °-774 0-787 0.800 

The following table shows examples of the practice of many 
of the leading Continental railways. In the table, 2r = diam- 
eter, / = length, <5 = thickness of hub, Q = total forcing 
pressure ; also, \V. I. ^ wrought iron, C. I. = cast iron, S = 
steel, C. S. = cast steel, B. S. = Bessemer steel. 



* See Procedings of the Royal Society, March, 
X., 1864, p. 238. 



1S73. Civilingenieur, Vol. 



* For an account of the strengthening of a piston head by shrinking ott 
bands, see the Berliner Verhandlungen, 1876, Sheet XVI. 

t Soxuetimes these surfaces are made slightly conical, such being the case 
in Nos. IS to 17 of the following examples. 



46 



THE CONSTRUCTOR. 



i 64. 
ExAMPi,ES OF Forced Connections. 



7 
8 

9 
10 
II 



12 
14 



15 
16 

17 



18 

19 
20 

21 
22 

23 

24 

25 
26 
27 
28 



29 

30 

31 
32 

33 
34 
35 
36 



37 
38 
39 



40 

41 
42 



43 
44 
45 
46 



47 
48 

49 
50 
51 
52 



DESCRIPTION. 



EASTERN RAILWAY OF PRUSSIA. 

Locomotive Driving and Coupled Wheels, 

Locomotive Trailing Wheels, 

Tender Wheels 

Car Wheels, with Spokes 

Steel Plate Wheels, for Cars, 

Steel Plate Wheels, for Locomotives, . . 

UPPER SII<ESIAN RAII,WAy. 

Locomotive Driving Wheels, 

Locomotive Trailing Wheels, 

Tender Wheels, 

Steel Plate Car Wheels 

Wrought Iron Spoked Car Wheels, . . . 

HANOVERIAN STATE RAILWAY. 

Locomotive Driving and Coupled Wheels, 

Locomotive Trailing Wheels, 

Standard Car Wheel, 

m,\gdeburg-h.ai<berstadt r. r. 

Car Wheels 

Car Wheels, 

Locomotive Wheels, 

S.AARBRUCK R.\ILWAY. 

Locomotive Driving and Coupled Wheels, 
Locomotive Driving and Coupled Wheels, 
Locomotive Driving and Coupled Wheels, 

Locomotive Trailing Wheels, 

Tender Wheels, 

Tender Wheels 

Tender Wheels, 

Standard Car Wheels, 

Freight Car Wheels 

Coal Car Wheels 

Passenger Car Wheels, 

RIGA-DtJNABERG RAILWAY. 

Driving and Coupled Wheels (Stephenson), 
Locomotive Trailing Wheels " 

Tender Wheels " 

Driving, Coupled & Trailing Wheels (Borsig), 

Tender Wheels (P.orsig), 

Passenger Car Wheels (Ashbury), .... 

Freight Car Wheels (Zypenl, 

Freight Car Wheels (Zypen) 

BORSIG LOCOMOTIVE WORKS- 

Locomotive Trailing and Tender Wheels, 
Locomotive Driving and Coupled Wheels, 
Crank Pins, 

WOHLER LOCOMOTIVE WORKS. 

Locomotive Driving and Coupled Wheels, 

Locomotive Trailing Wheels, 

Tender Wheels, 

NORTHERN RAILWAY OF FRANCE. 

Locomotive Wheels (Stephenson), . . . 
Locomotive Wheels (Clapeyron), . . -. . 
Tender Wheels, with strong hubs, . . . 
Crank Pins, 



PARIS-LYONS-MEDITERRANEAN R. R. 

Locomotive Driving Wheels, 

Locomotive Trailing Wheels 

Tender Wheels, 

Car Wheels 

Car Wheels, 

Crank Pins, , . . . . 



Dimensions. 
Inches. 



7'A 
VA 

sVs 

1% 



TA 

rli 

on; 

5^ 



7'A 
5>s 



5>s 
5>s 



7 
7 

7H 
6 

5% 

cl5 
Ol 6 

5% 
5% 
5% 



7 

5 'A 

(>A 

c 9 
OTti 

aU 
s'A 



6to8 
6|-8 
4to6 



7'A 
7A 

5A 



7A 

TA 
7 



6H 

7A 
7-h 
6U 
7i\ 



7 
614' 



7A 

6J-i 



7% 
8 

/IS 

6 

7fs 

OTiJ 

7'A 

&A 

7% 
7A 



7A 
(>/z 
7 

S/s 
7 

6/2 
loX 

SA 



6J-7 
7to8 
7toS 



7A 
6U 



,15 

3 

2A 
2^ 
2 

2lV 



3% 
3A 
3/s 

■"16 

2A 



3^ 

2A 



A'A 
3X 



2^ 

314' 

2fl 

% 

2X 
2 

2tJ 

4>s 

2A 

3/s 



3% 

3 
3X 



2i-3 

3to4 

-2} 



3^ 
3 

2A 



Material- 



Hub. Axle 



W.I. 
W- I- 
W. I. 
W. I. 

cs. 

C-S. 



W- I- 
W. I. 
W. I. 
CS- 
W- I. 



W. L 
W. I. 
W. I. 



W.I. 

cs. 

W. L 



C.I. 

W. I. 
W. I. 
W. I. 
C. I. 
W. I. 
W. I. 
W. I. 
C. I. 
W. I. 
C. I. 



W. I. 
W. I. 
W.I. 
W. 1. 
W. I. 
W. I. 
W. I- 
W- I. 



W. I. 
W. I. 
W. I. 



W. I. 
W. L 
W- I. 



W.I. 
W. I. 



W. I. 
W. I- 
C.S- 
W- I- 



C.S. 
CS- 
C-S- 
C-S. 

cs. 

C-S. 



CS- 
C-S. 
C-S. 



CS. 
CS- 

cs. 

CS. 
W. L 
W.I. 
CS. 

cs. 

W. I. 
W- I- 
W. I. 



W. I. 
W. I. 
W. I- 
W- I- 
W- I. 
W. I. 
W. I. 
B-S- 



Q 
Pounds- 



160,600 
160,600 
118,800 
110,000 
110,000 
154,000 



220,000 to 330,000 
165,000 to 220,000 
iio,oooto 132,000 
110,000 to 132,000 
132,000 to 154,000 



165,000 to 176,000 
143,000 to 154,000 
88,000 to 110,000 



110,000 to 132,000 
1 10,000 to 132,000 
1 76,000 to 198,000 



137,940 
247,588 
247,588 

198,000 

136,840 

i6S,o8o 

198,000 
165,000 to 193,600 

165,000 

193,600 
II 0,000 to 165,000 



90, 200 
go, 200 
85,800 
90, 200 
85,800 
68,200 
77,000 
88,000 



W.l.orS. 155,000 to 220,000 
W.l.orS. 220,000 to 330,000 
W.l.orS, iiOjOOOto 165,000 



W.l.orS, 
W.I,.orS 
W.l.orS 



W. I- 
W- I. 



W- I. 



W.l.orS, 

W.l.orS, 

CS. 



220,000 
220,000 
132,000 



176,000 

176,000 

176,000 

33,000 



77,000 to 88,000 
55,000 to 66,000 
55,000 to 66,000 
39,600 to 48,400 

22,000 

66,000 



REMARKS. 



With Key. Data as given. 
No Key. Data furnished. 
No Key. Data furnished. 
No Key- Data furnished- 
No Key- Data furnished. 
No Key. Data furnished. 



With Key. Measured Dimensions. 
With Key. Measured Dimensions. 
No Key. Measured Dimensions. 
No Key. Measured Dimensions. 
No Key. Measured Dimensions. 



With Key. Data furnished. 
Without Key. Data furnished. 
Without Key. Data furnished. 



No Key. Measured Dimensions. 
No Key. Measured Dimensions. 
No Key. Measured Dimensions. 



With Key. 
With Key. 
With Key. 
No Key. 
No Key, 
No Key, 
No Key. 
No Key. 
No Key 
No Key, 
No Key. 



With Ke}'. 
With Key. 
With Key. 
With Key. 
With Key. 
With Key 
No Key. 
No Key. 



Measured Dimensions. 

Measured Dimensions. 

Measured Dimensions. 
Measured Dimensions. 
Measured Dimensions. 
Measured Dimensions. 
Measui-ed Dimensions. 
Measured Dimensions. 
Measured Dimensions. 
Measured Dimensions. 
Measured Dimensions. 



Measured Dimensions. 

Measured Dimensions. 

Measured Dimensions. 

Measured Dimensions. 

Measured Dimensions. 

Measured Dimensions. 
Measured Dimensions. 
Measured Dimensions. 



With Key. 
With Key. 
No Key. 



Measured. 
Measured. 



With Key. Measured- 
No Key. Measured. 
No Key. Measured. 



Data furnished. 
Data furnished. 
Data furnished. 
Data furnished. 



With Key. 
No Key. 
No Key. 
No Key. 
No Key. 
No Key. 



THE CONSTRUCTOR. 



47 



From example No. 12 we obtain in formula (63) the value 
5 X 176,000 



S, = \ 



= 5,336 lbs. 



7-5X^X7 

According to (65) ^ = 0.53, and substituting these values in (64) 
gives ^"2 = 10,679 ll^s. 

From example No. 10 we have : 
5 X 132,000 



hence a = 3.86" X 0.92 = 3.5s" 

The actual thickness of the hub -was 3-54"- 

The ring form is not the only form of construction -which 
may be used for joining members by forcing, since other forms 
may also be used. An example may be found in Erhardt's 
flange joint, Fig. 17S* In this case clamps of hardened steel 



^1 = 



5.125 X ^ X 7-3125 

also p = 0.44, and hence S^ = 12,734. lbs. 
From example No. 37 we have : 



= 5,603 lbs. 



9 — — 
'-'1 — ' -. 



5 X 220,000 



; 6,970 lbs. 



7-5 X - X 6.7 
also p = 0.526, giving S^ = 13,250 lbs. 

From eyample No. 16, taking Q= 132,000 lbs., we get S^ = 
4867 lbs.; /J =0.77; S.i = 6i2o lbs.; and in No. 17, we have i'j ^ 
,6617 lbs.; p = 0.569, and52:= 11,629 lbs., neither of which are 
excessive. 

The force required to force a hub off an axle upon which it 
has been pressed, is not materially different from the force with 
which it was pushed on. The bore of such a hub may also be 
reduced when necessary by forcing rings upon it. Such rings, 
when used for car wheel hubs, are usually made of rectangular 
cross sections, the diameter ranging from 2"Xi", to i^'^X 
iX", etc. 

An inspection of the table will show that there is a tendency 
towards increasing pressures. For car wheels, where until quite 
recently, pressures of 60,000 to 90,000 pounds were used, we 
now find So,ooo to 110,000 pounds not infrequently; while for 
locomotive wheels, over 200,000 pounds is the rule. 

Midway between the methods of shrinkage, and of cold forc- 
ing comes the lesser used method of expansion by use of boiling 
water. This system secures a much more uniform action of the 
temperature than is practicable with a red heat, and has been 
used with excellent results upon the Russian railways for fitting 
tires to plate wheels. The tires are suspended by a crane, in a 
tank of water which is kept at the boiling temperature by a jet 
of steam (the allowance for expansion being a little less than 
-|-Jjf of an inch to the foot of diameter. An immersion of 10 to 
15 minutes is required to obtain the desired expansion. Three 
workmen can in this manner fit 12 to 14 tires per day of eleven 
hours. This method may also be found applicable to the fitting 
of hubs. 

65. 
Dimensions of Rings for Cold Forcing. 

Since the forms of the various hubs may be taken as cjdin- 
drical in nearly every case, the stress^niay be calculated by the 
formulaj already given. Itis, however, desirable to present these 
in such form that they may be used to determine the thickness 
of hub which, when forced on cold, shall resist a determinate force. 
In (62) instead of the radial stress 5'j, substitute the tangential 
stress S.,, giving = 2 - y'/y^Sj p, which combined with (65) 
gives : 



4=Vs 



rl/S,+ Q 



In this, Q\b the maximum force w'hich the hub can oppose to 
turning, at the diameter of the fit. If we take the moment 
of the force tending to rotate the wheel as P R^ we must have 

Q ?^> PR' ^Y> """i^^ then be the factor of resistance against 

slipping in any such case. This mode of attachment is then 
only practicable when 2 tt 7'1/S.> > O. By choosing different 
values for 6*2, and Q, various thicknesses for the metal of the 
hub may be obtained. 

Example. The following data are taken from Borsig's Express Locomo- 
tive at the Vienna Exposition ; Two pairs of coupled driving wheels of 
38.19" radius, without keys; bore of cjiinders 17 ; steam pressure 147 
pounds ; crank radius J? = 10". If we suppose the entire force upon the 
i)iston to act upon a single wheel, we have : 



PJ?^ (17)2 X 07S54 X 147 X 10 = 33,366 X 10 
The bore of the wheel is 7.72" hence r = 3.S6'' while / == 7-S7". 
P R 333,660 



This gives 



3.S6 



■ = 86,440 lbs. 



The moment 333,660 is that which the friction of the wheel upon the axle 
should be able to resist without slipping. Hence it follows that Q must 
neessarily be greater than 86,440. If we take a value of iQ = 154,000 lbs-, 
thus giving ample margin against slipping, and also use a wrought iron 
hub, making S2 = 7120 lbs., taking y"= 0.2 as before : 



/• 



2 TT X 3-S6 X 7.87 X 0.2 X 7T20 + 154,000 
2 IT X 3-86 X 7.87 X 0.2 X 71=0 — 154,000 




Fig. 17S. 

are used to create the pressure. These clamps serve to press 
the light flanges together, and they may be forced on by use of 
a screw clamp or other suitable press. Tests of such joints 
under steam, pneumatic and hydraulic pressure have shown the 
joint to be tight and serviceable. 

The system of forced connections has grown into extensive 
use, and appears to be applicable to many forms of construc- 
tion, and it is to be hoped that the forcing press, for which the 
firm of Schaeffer & Budenberg have made suitable pressure 
gauges, may be found an indispensable tool in all large 
workshops. 

CHAPTER III. 

KE VING. 

'i 66. 

Keyed Connections. 

The simplest form of keyed connection consists of three 
parts, viz. : the two parts to be connected, and the key itself. 
The key is made with a slight amount of bevel on both sides, 
or a greater angle on one side, according to the manner in 





which the connection is made. The trigonometrical tangent 
of this angle is called the draft of the key. In Figs. 179 and 
180 are shown both forms of keys. For the latter form we will 
assume that both sides have the same angle. 

Let: 

o := the angle of draft, 
P^the force to be transmitted, 
Q = the driving force upon the key, normal to P, 
0''= the opposing force, tending to drive out the key, 
y=/o-(j, the co-efiicient of friction between the surfaces 
of the three parts. 
For ke3-s with draft upon one side, we have : 
= P2tg(a-\-2<i) \ 
Q'=P2ig(2^— a) i 

In order that Q' should not be negative and the key come out 
of itself, we must have a< 2 <p. 'Porf= O. 1, this gives t g a < ^-. 



(67) 



117,656 



: 0.92 



* Royal Prussian Patent, May 23, iS-6. No. 159. 
model and description. 



Illustrated by drawing, 



48 



THE CONSTRUCTOR. 



For keys with draft on both sides we have approximately : 

\ (6S). 

In this case it is necessary to keep below the full value ofy, for 
each edge of the key in order that the connection may not force 
itself apart. The total draft will be found to have approximate- 
ly the same minimum value as in the previous case. 

In practice it has been found that keys which have shown en- 
durance and resistance under load, have been made with a total 
draft of yV, i; J, and even j-J ^ or less, while others made with Jj, 
j'j or sometimes Yf, are less secure. 

The load upon a key may act in three different manners each 
of which may again be positive or negative. In the first, the 




Fig. iSi. 

load acts normal to the base of the wedge, as at QK, Fig. iSi, 
or as P, iu Fig, 179 and iSo ; and for this form, the term Cross 
Key may be used. The second case occurs when the load acts 
normal to the plane K H Q, as K L, iu Fig. iSi, which may be 
called a Longitudinal Key. The third case is that in which the 
force acts normal to the plane Q K L, as Jv H, Fig. iSi, which 
may be called a wedge key. 

? 67. 

Cross Keyed Connections. 

In Fig. 1S2 we have an example of a cross keyed connection. 
The rod and the key are both of wrought iron, the boss is cast 
iron. The stresses for a given force P upon the rod are : the 
bending stress upon the key, as in Case VIII. § (6) (Stress 5,) ; 
the shearing stress between the key and the inner edge of the 




Fig. 1S5. 



boss (Stress S^), and the tension upon the segment shaped sec- 
tions of the rod on both sides of the mortise for the key (Stress 
^'3). If, according to § 2, we make S.^ = o.S S^, and ^i = S^, we 
have : 



h-- 



■ (' 



b = 0.267 ^> or say 



(69). 



If we make /;j=o.S d, h.^ = d, c!=: 0.5 d, we shall have good 
practical proportions. Iu Fig. 1S3 we have two wrought iron 
rods coupled by wrought iron keys. In this case a wrought 
iron sleeve is used, whose thickness ^ = 0.25 d. Fig. 1S4 shows 
a form similar to Fig. 1S2, except that the key passes below the 
boss, instead of going through it, while in Fig. 1S5 the key is 
let into the side of the rod. 

The pressure upon the base surface of the key in the case of 
Fig. I S2 may be taken as : 

P^ _ (0.7854 ^'-^«^) ^3 
' b d bd 

which gives / ^= 2.14 S^ (70) 

quite a high enough value, especially if we take, in Fig. 1S3, 
0= 0.25 d. The pressure becomes yet higher for the method 



shown in Pig. 1S5, for which case the value of S^ shonld not be 
taken too great. If the connection is intended to be taken 
apart frequently, the value of p should not be allowed to be too 
great. This may be accomplished either by reducing the value 
of .Sj, or by providing an increased cross section of metal about 
the mortise for the key, or by extending the surface by means 
of cotters or gibs, as shown in Fig. 1S6. The key may then be 
made smaller than already given above. The forms of keyed 
connection shown are used for example in the rod connections 
of water wheels, and in similar cases. 





Fig. 1S6. 



Fig. 187. 



In Fig. 1S7 is shown a method of keying a foundation bolt. 
The gibs or cotters are used to increase the strength. Fol- 
lowing the calciilations of (; 12, the depth of the three pieces- 
may be made alike in the middle. The anchor plate in the 
foundation masourj' should be arranged so as to give access to 
a nut on the lower end of the bolt, and this can be tightened 
by hand until the bearing is thrown upon the key, and the 
driving iu of the latter binds all the parts firmly together. 

?6S. 

LoNGiTUDiNAi^ Keys. 

Keys of this class are principally used to secure the hubs of 
wheels to their shafts or axles. For this purpose they may be 
considered as divided into three classes, as follows : 
Concave, or hollowed keys. Fig. 18S, 1 ; 
Flat Surface keys. Fig. 188, 2, 4, S, 
and Recessed keys, Fig. 18S, 3. 
The Concave key is only suitable for constructions involving 
small resistance, and acts merely bj' the friction due to the 
pressure which it causes. The flat surface key is capable of 



^^ .&-«. Jzi-y. ^^^^ 





Fig. 1 88. 

resisting much greater force and vibration, and when used in 
the multiple manner shown in 4 and 5, it makes a secure and 
efficient fastening. The recessed key, shown in 3, affords a 
very secure method of fastening hubs to shafts to which they 
have been closely fitted, and is simply and readily made. Ke5'S 
of this kind are also used as an additional precautionary fasten- 
ing for hubs which have been forced on. 

In determining the dimensions of keys it will be found most 
convenient to use empirical methods, except iu cases of great 
vibration ; the following formulse will be found to cover the 
usual range of work. The material for the key is taken as 
steel, and distinction is also made between cases in which the 
hub is subjected merely to endlong pressure, and those where 
torsional stresses exist. The former may be called draft-keys, 
the latter torsion ke3'S. 

If we call the diameter of the shaft D, the breadth of the 
key S, and the middle depth of the key S-^^ we have : 

for Draft keys, 5= 0.24" ■\ ; 5, = 0.16" -| 

^ D D 

for Torsion keys, S ^ 0.16" -|- ; 6'i = 0.16' -| — — 

The taper of such keys is made about xij 



(71) 



THE CONSTRUCTOR. 



49 



For the more commonly occurring diameters we have the 
following proportions : 
£> = !>' 2" 3" 4" 5" 6" 7* S" 9" 10" 

For Draft Keys : 



c a// 

•-> "8 

Q" — 1// 






A'' 



_9_// S// 



a// 



T.¥ 



A// 



a// 

4 

X" 



9 // 



For Torsion Keys. 

3 // Til// J 9 // 

S// Z// 
16 4 B 

diameter than 1 ' 






4 -^ "^S 

// T_l_ tJL// 

■'is ■'lu 
we mav make 



c 1// 

•-"l 4 C Z IS 

For shafts of less 

5'= , S^-= If several keys are to be used, they maj' 

3 5 

be made the same dimensions as single keys. For hubs which 
have been forced on, and hence would be secure without any 
kc}', the dimensions for draft keys may be used. 

?69. 

Edge Keys. 

When the pressure upon a key acts at right angles to the 
plane oi its height, the difference between the positive and 
negative direction of the forces is readily distinguishable. 



1±-- 




Fig. 189. 



When the pressure acts as in Fig. 189, the combination is inse- 
cure, since the only binding action of the key is that due to the 
pressure, and consequ .*nt friction between the parts. If the 
base of the key is ro agh, and the inclined face smooth, the 
action of a force in the direction H' , tends to tighten the parts 
together. An application of this action is shown in the curved 
key of Keruaul, shown in Fig. 190. When the hub is rotated 




Fig. 193. 



Fig. 192. 

in the direction of the arrow, the action is the same as that of 
the force H' , in the preceding case, and the shaft is firmly 
grasped. A countersunk screw at a, is used to tighten the ke)', 
and a similar one at b, to loosen it. This principle will be dis- 
cussed later, under the subject of couplings. 



\ 70. 

Methods of Keying Screw Propei.lers. 

In securing the propellers of steamships the greatest care 
must be observed in the methods employed, and in their appli- 
cation. In Fig. 191 is shown Rennie's method of securing one 
of the blades of a Griffith's double bladed propeller. In this 
case a rectangular key is used, passing through a cjdindrical pin 
which is cast in one piece with the blade and which is in turn 
held firmly by the four smaller keys shown. These latter keys 
are held in their places by caps secured firmly by jam nuts. (See 
^71.) The blade and hub are both of bronze. 

Example. In a propeller by Penn & Son, d = 15", ?»= 7>^", i = 2K". 

Fig. 192. This shows a method used by Maudslay, Sons and 
Field, Raveuhill & Hodgson, and others. Two rectangular 




Fig. 191. 

keys, passing through the hub of the propeller boss, and re- 
cessed into the metal of the shaft, act at the same time to receive 
the thrust of the screw and to prevent rotation upon the shaft. 
In this case the hub is made of bronze. 

Example. In the *' Lord Warden," the middle dianjeter Qfd= 19", /^52", 
/i= SJ^", d= 3^8" ; in the " Lord Clyde," d ^205^5", ^= y/^, h =^ 10", ^^3". 

Fig. 193. This shows a method of using two longitudinal 
keys. The hub is bored with a quick taper, and a heavj' bronze 
nut holds the hub upon the cone, while the longitudinal keys 
resist the action of torsion. 



Example. In the "Minotaur,' 
ter <i = iSJi", /=4S", 3'= 3'. 



en^ned by Penn & Son, the mean diame- 



The ordinary rectangular keys are also used to secure screw 
propellers, as well as special forms of fastenings.* 

I IT- 

Unloaded Keys. 

The force P, which under ordinary conditions bears upon a 
key, may by various methods be supported by other means ; 
the key in such a case may be said to be unloaded. Such con- 
structions offer a much greater security, and permit much 
lighter keys, than the methods previously described. A few 
examples will serve to illustrate. 



= See N. p. Burgh, Modem Screw Propulsion, London, 1S69. 



so 



THE CONSTRUCTOR. 



Fig. 194. This shows a form of counection used for mine 
pump rods; the interlocked uotches receive the load of tension 
upon the rod, and the hollow key only serves to bind the parts 
together without itself supporting any of the weight. Fig. 
195 shows Wiedeubruck's rod connection.* The hub is made 




F:g. ig.1. 



Fig. 195. 



Fig. 196. 



In 

by 



in halves and the reversed conical seats receive the load 
Fig. 196 is showu a connection for two intersecting plates 
Bayliss.t 

The method of keying shown in Fig. 192, H, may be made 
quite secure by relieving the key from the load, and examples 
of this form are often found. 

Methods of Securing Keys. 

In order that a key may not back out under its load, the 
angle of taper should be less than ^%, or if it is symmetrical in 
form, each side should be less than ^L, providing the co-efS- 
cient of friction is equal to j^. Even when the taper is made 






Fig. 197. Fig. 198. 



Fig. 199. 



Fig. 200. 



Fig. 201. 



less than this, however, keys are very apt to become loose if 
they are subjected to much vibration, and to sudden and irregu- 
lar changes of load. In order to provide against such emer- 
gencies, and also in order to permit the use of greater taper, 
various methods of securing keys are emploj'ed. 

The simplest method consists in splitting and spreading the 
small end of the key, and for some purposes this is sufficient. 
la order to prevent a key from flying back, or jumping out, the 
projecting end may be drilled and fitted with a split pin. For 
the keys used in connecting rod ends various methods are used, 
examples of which will be found ki the following figures. Figs. 
197, 198, and 199 use screws under tension. When these are 
used in locomotives or marine en- 
gines, the screw is again secured by 
the use of a jam-nut. Fig. 200 is 
used with a set-screw, the point of 
which bears in a shallow channel in 
the side of the key, so that if the 
pressure of the set-screw is unable 
to hold the key, it will at least keep 
it from flying out. The channel is 
also of service in preventing the 
point of the set-screw from marring 
the finished surface of the key. Fig. 
201 shows a form of screw-clamp. 
This clamps the key by drawing the two blocks tightly against 
its sides ; the screw passes through a slot in the key. 

Fig. 202 shows the method employed in securing the key 
used in the form of fastening for screw propellers, used by 
Maudslay, and shown in Fig. 192. A small block is bolted fast 
to the projecting end of the key, and a bronze cap is screwed 
down over all. 




Fig. 202. 



* German Patent, No. 1507. See also patent No. 510 of H. Rademacher for 
improved rod connections. 
+ See Pract. Mech. Journal, Vol. Ill, 3d Series. P. 342. 



CHAPTER IV, 

BOLTS AND SCREWS. 

I 73- 

Geometrical Construction oe the Screw Thread. 

Screws are used in machine construction to produce three 
kinds of effects, viz.: for clamping or joining parts together, 
for producing pressure, and for the transmission of motion. We 
shall now only consider the first two classes. Screws may be 
classified with regard to the shape of the cross-section of their 
threads into : 

Triangular or V, 

Square or Rectangular, 

and Trapezoidal. 

All these forms belong practically to the so-called axial screw 
thread surfaces.* By this is meant the surface which is described 
by a right line ABC, Fig. 203, when one of its points remains 
upon a directrix, in this case the axis O D, of the screw, while 
the generating line itself maintains a constant angle a, with the 
axis, proportional to the advance which the directing point 
makes u-jon the axis. The angle a is called the angle of 
advance, and its complement B, is the base angle of the screw 
thread. These are either V, or square, according as the angle a 






Fig. 203. 

is an acute or right angle. The normal cylinder upon the axis 
O D, upon which the screw thread is described, is called the 
pitch cylinder. This cylinder is supposed to pass through the 
threads of the screw at such a point that two adjoining sections 
bear the relation of screw and nut. 

_ The space passed over by the directing point during one rota- 
tion around the axis is called the pitch of the screw, and will 
hereafter be designated by the letter ,5 ; and the angle which a 
line tangent to the screw-thread at any point makes with the 
base of the pitch cylinder, is called the pitch angle, called f*. 
From this it will be seen that threads described upon concentric 
cylinders may have the same pitch, but different pitch angles. 
The area of a V screw thread may be taken as equal to the 
sum of the surfaces the two halves of the thread, opened out to 
an angle of 180°. For rectangular threads the area is simply 
that of the corresponding simple surfaces. For trapezoidal 
threads the area is equal to the sum of the inclined and parallel 
surfaces (see \ 86). 

I 74. 

Dimensions of V Screw Threads. 

For any given force P, acting parallel to the axis of a screw, 
the resistance of the metal of the body of the screw may be 

determined according to Case I., page , but the stress may 

instead be taken as a simple case of tension if the value of S, 
be not made too great. If we take for wrought iron, 6'= 3600, 
and let d^ be the diameter at the base of the thread, we have : 



fl'j = 0.02 •%/ P 
P=275odi 



(72) 



The nut is generally hexagonal, but is sometimes made square ; 
we will here limit ourselves to the former shape. The thickness 
of the nut is usually made equal to the outside diameter d, of 
the screw. This makes it much stronger than the threads of the 
screw ; f but the depth is desirable, as it distributes the pressure 
over a greater area of screw thread. For the superficial pres- 



*See the article : Concerning some Properties of Screw Threads. Berliner 
Verhandliiiig, 1878, p, 16. 

t Recent investigations made at the Stevens Institute at Hoboken, show 
that the resistance of the thread is reached when the thickness of the nut 
-s made 0.45 to 0,4 d. See Railroad Gazette, 1S77, November, p. 4S3, 



THE CONSTRUCTOR. 



SI 



35ure/i, we have for a depth of thread / and ?; threads iu the nut, 
both for V, and square threads : 



i> = 



; pitch S, and making n 



_^ 

Introducing the pitch S, and making ns = d, we have : 
4 ■ i 



(73) 



(74) 



In both equations the third member may be neglected.* 
The value of p should not be permitted to exceed 1440 lbs. 

and -J- : 



If « = 8, 



12, we have, taking 5', as above, 

p := 3600 X I (i — T + tit) or about = 1000 lbs. 

In the consideration of this subject, the friction of a screw 
should not be neglected. 
Let: 

Q = the force acting at the mean diameter of a screw, 

normal to the plane of the axis ; 
6' = the pitch angle of the thread at the mean diameter, 
/=: ig <p, the coefficient of friction ; 

■we then have, in order to overcome the thread friction, due to 
the force P, for square thread ; 

or for the resistance : '■ (75) 

f-ts6' 



Q = P 



■■Ptg {<?+&') 



■while for V threads, we have : 
f'±tg6' _ 



Q = P- 



-'f'tgi' 



■Pig{'P — ^') 



Ptg (,■/ ± <50 



•(76) 



in which /' = 



/ 



cos (3 



In order to overcome the friction on the base of the nut also, 
the value of Q must be more than twice as great. For ig S' , 
we may take then tg &. This value is usuall}- so small that the 
friction often cannot resist the load, and the value of O' becomes 
negative. 

I 75- 

The Whitworth Screw System. 

By a system of screw threads is meant a collection of rules or 
■formulse by which the profile of thread, pitch, diameter, and 
■other details of screws and nuts may be determined. Such a 
system was first formulated by Whitworth in 1841, and since 
that time the subject has been more and more studied, until it 
is now considered one of the greatest importance,! especially in 
regard to the metric system. 



»-I 




A united opinion on the subject has not yet been reached. 
Many weighty reasons have been advanced for the introduction 



* P=pir t id — t) n, hence p = S d^- : tth — 

~—~ I — ~ — 1 : I -3- , which, by neglecting I 



.S I 

• 

4 " 
+ , etc., 



-gives the above result. In this case, p is the pressure upon the projected 
area of the screw thread. 

t See The Metrical Screw System, etc., by a Committee of the Society ot 
-German Engineers. Berlin, Gartner, 1S76. 



of the Whitworth system into Germany, while others, equally 
strong, have been advanced for the metric system. 

The Whitworth system takes for the form of a section of a 
screw thread an isosceles triangle whose base is equal to the 
pitch s, and whose angle at the apex = 55°, from which the 
height /„ = 0-96 .y. The thread is rounded at top and bottom to 
an amount equal to ^l ^0, so that the working depth i^-f^ /„ ^ 
0.64 s. The values of the pitch .s were given by Whitworth in 
a table which extended to i'^.- The use of this system developed 
some deficiencies, among others the difficulty of originating the 
cross-section of thread, and the gradation of diameters. The 
original table of diameters was not altogether satisfactory to 
Whitworth himself, and in 1S57 he extended the old table by a 
new one, which, since that time, has been known in England 
as the standard system for bolts and nuts.f In Germany, how- 
ever, the whole subject is yet under active discussion. 

The following table gives the old and new scales, the values 
of d and j being in English inches. The values for jY' ^^^ 
J-^" are only given approximately. 



Whitworth's Screw TitRE.A.D 


Scales. 




New Scale. 


Old Scale 


1 


New Scale. 


Old Scale. 


I 


d. 


d. 


5. 


d. 


d. 


5. 


O.IOO 




48 


1.I2S 


1% 


7 


0.125 


% 


40 


1.250 


^K 


7 


0.150 




32 


1-375 


^y% 


6 


0-I75 




24 


1.500 


^'A 


6 


0.200 




■ 24 


1.625 


iy% 


5 


0.225 




24 


I- 750 


^Ya, 


5 


0.250 


% 


20 


1-875 


T-Vi 


4K 


0-275 




20 


2.000 


2 


AA 


0.300 


A 


18 


2.125 


2% 


aA 


0-325 




18 


2.250 


2% 


4 


0-350 




18 


2.375 


2^ 


4 


0-375 


'A 


16 


2.500 


2K 


4 


0.400 




16 


2675 


2% 


4 


0425 




14 


2.750 


2Y 


3K 


0450 


-h 


14 


2-875 


2A 


iA 


0-475 




14 


3.000 


3 


iA 


0.500 


% 


12 


3250 


3X 


2A 


0-525 




12 


3-500 


3K 


3A 


0.550 


• 


12 


3-750 


3^ 


3 


0-575 




12 


4.000 


4 


3 


0.600 




12 


4.250 


A% 


2ji 


0.625 


'A 


II 


4.500 


aA 


2rs 


0.650 




II 


4-750 


4^ 


23/ 


0-675 




II 


5.000 


5 


2^ 


0.700 




II 


5-250 


^% 


2A 


0-750 


H 


10 


5-500 


5A 


2% 


0.800 




10 


5-750 


5K 


2A 


0.S75 


% 


9 


6.000 


6 


2A 


0.900 




9 






1. 000 


1 


8 







Whitworth's Pipe Thre.-vd Scai,e. 



d=y^ 


A 


H 


A 


H 


I 


iX 


^A 


i}( 


2 


n = 2& 


19 


19 


14 


14 


II 


11 


II 


II 


II 



The regularity of the progression might be improved upon. 
This may be more clearly illustrated in the following diagrams- 
The greatest irregularity lies between the sizes from 3_j''/ (-q 
252" ; and the gradation of diameters is also uneven. The 
cause for this lies in the system of measurement used. Whit- 
worth evidently perceived the desirability of introducing a 
decimal notation, but also wished to retain the fractional divi- 
sions in halves, quarters, eighths, &c. ; this has partly been 
secured, neglecting sixteenths, by having the gradation based 
upon fortieths, and their combinations as shown in Fig. 206. 

For the pressure p, we have from (74), taking t = 0.64 5 : 



P 



4 X 0.64 



[- 



1-9 ^r + 0-4 



(^)' 



If we make S = 
the values of p ■ 
have, when d = 

0.02I2. 



= 3555 lbs. we have for </= O-i", 3", and 6", 
= 93S lbs., 1152 lbs. and 1209 lbs. For t g d, we 
o.i'^, 3", and 6", the values 0.0663, 0.0303, and 



* Briggs stated the relation of pitch and diameter of the Whitworth system 
to be approximately : — 

s = 0.1075 d — 0.0075 ^' -f 0.024. 

fSee -£"«^. aud Arch. Jouryial, 1S57, P- 262 ; 1S58, p. 4S ; also Shelly, Work- 
shop Appliances. Eondon, 1876, p. 102. 



52 



THE CONSTRUCTOR. 




d = _5_ Ji. J_5 ^ ^ JQ. ^ ii 
10 lO dO 40 10 40 -10 10 

Fig. 2o6. 



2'/ "'^ 




2*/^ 


v^ 


^ 


- Ay' 





















^\y 


























4 /' 
























^' 


























V- 


























'/ri i 


























«(^ 


' 1 


























.)0 1 


? ■ 


























y^- 




\ 'i 


























,20 






































*■ 



























a— 



Fig- 205. 



? 76. 

SelIvErs' Screw Thread System. 

The confusion in the use of screw threads having become 
■very troublesome in the United States, Mr. William Sellers 
brought before the Franklin Institute, in 1864, a system which 
he proposed for general use.* A committee of the Institute 
reported upon the system in December of the same year, and 
recommended its general adoption, f This system is now 




Fig, 207. 

generally known as the Sellers System. The profile of this 
thread is shown in Fig. 207. The thread angle 2 /3 = 60° ; the 
depth i = 0.75 /^ = 0.65 j>. The pitch is determined by the 
formula 5'= 0.24 y/d + 0.625 — 0.175, the result, as with Whit- 
worth's system, being so modified that the number of threads 
per inch shall be a whole number. 

The following table gives the adopted number of threads per 
inch for various diameters : 



rf=X 


l\ 


Vs 


A 


% 


t's 


% 


V^ 'A 


i = 20 


18 


16 


14 


13 


12 


II 


10 9 


d=i/s 


iX 


i^ 


iX 


1% 


m 


i^ 


2 


T = ^ 


7 


6 


6 


5% 


5 


5 


A% 


d=2'X 


^% 


ni 


3 


Z% 


3K 


31^ 


4 


^=.% 


4 


4 


i% 


lY^ 


3X 


3 


3 



d = 5 5)4 5'A 5'/ 6 

— = 2yi 2% 2^ 2^ 2X 
.J 

The Sellers System compares very favorably with the Whit- 
worth System, and notwithstanding the difference in profile, it 
gives almost the same depth of thread. The angle is very con- 
venient, and the simplicity of the profile is such that a suitable 
tool may easily be made and used in the shop. These facts 
explain the rapid introduction of the system in America. The 
progression of the pitch is also more uniform than in Whit- 
v^forth's System ; and the uncertainty about the thread of the 
Yz" screw, which was always a stumbling block in the original 
Whitworth Scale, is avoided. The values for ^-^" and ■^" are 
retained as in the Whitworth Scale of 1S57, and y\" is also pro- 
vided for, so that the requirements of the English system of 
measurements are fully met, up to 2" . 



§ 77- 
Metricai, Screw Systems. 

Recognizing the advantages which have followed from the 
introduction of the Whitworth System, various attempts have 
been made to devise a system of screw-threads which shall be 
adapted to the metric system of measurements. The following 
fourteen systems have been suggested : 

Armengaud, Redtenbacher, Paris-Lyons-Mediterranean R. R., 
Northern Railwajr of France, J. F. Cail, the French Navy^ 
Bodmer, two systems proposed by Ducommun, of Mulhouse. 
Alsace ; the Engineering Society of Mulhouse, Reishauer & 
Bluntschli, of Zurich ; the Pfalz-Saarbriick Society of German 
Engineers, and two systems of Delisle. 

The formula; and tables given in the previous editions of this 
work have also been spoken of as systems, but they are not en- 
titled to any such position, as they were merely adaptations of 
the Whitworth system. The number of proposed systems may 
be taken as an indication of the difficulty of the task. Indeed, 
it is only by very carefully weighing the respective merits of 
the various plans, that it is possible to say which is the best. 
The following requirements should be kept in mind as essentials 
in considering any system : 

1. The profile of the thread should he such as viay be readily made nnih 
requisite accuracy. In this respect Whitworth's system is deficient, and 
the profile of the Sellers thread is to be preferred. 

2. The pitch should be given, so far as possible, directly by the formula^ 
without requiring any vtodif cation of its resi4lts. Both Sellers and Whit- 
worth are deficient in this point, since they both modify the results of their 
formula.* 

3. The gradation of bolt diameters should be so disposed that fractions of 
niillitnetei'S sitould not occur in diameters, afid that their gradation should. 
conflict as little as possible with the decimal system. 

All three of these requirements should be attained within the 
limits of generally used sizes, and should at least extend to 
bolts of 80 mm. in diameter. The last three systems, viz. : the 
Pfalz-Saarbriick sj'stem and the two of Delisle, are the only 
ones which appear to have considered these points, and these 
we shall examine somewhat in detail. 

?78. 

Metrical Screw Thread Systems. 

DEMStE /, Pf.\i,z-Saarbruck and DeuslE //. 

The following three diagrams show the gradation of pitch and 
diameter for the three systems, the ordinates representing the 
pitch being shown on five times the scale of the bolt diameters, 
and the values being also given for d and j in the annexed tables. 
In the first two cases the profile of thread is exactly the same as 
in Sellers' system, while in the third, the base angle is made 
26° 34'. This has been chosen for the purpose of making the 
theoretical height of the triangle of the thread equal to s. The 
thread is flattened as in the Sellers system. 

All three of these systems are marked by simplicity and in- 
telligibility. These features have been attained by abandoning 
the idea of representing the relation of 5 to rf by a single equa- 
tion (such as that of a parabola), and using two or more equa- 
tions of straight lines. A noticeable irregularity exists in the 
Pfalz-Saarbriick system between the diameters of 26 and 28 mm., 
indicating that a somewhat finer pitch is used in proportion to 
the diameter below 26 mm. 

The second system of Delisle is rather simpler than the first ; 
there is also an important difference in the angle of thread, as 
will be seen subsequently. 



* Journal of the Franklin Institute, 1864, Vol. 47, p. 344. 
f Journal of the Franklin Institute, 1S65, Vol. 49, p. 53. 



*In the old Whitworth scale all 33 values were modified ; in the Sellers sys- 
tem this is done with 31 out of 34 sizes. 



THE CONSTRUCTOR. 



S3 




Deusi,e /. Fig. 20S. 



d= 4 5 6 j 7 


8 jio 12 !i4 


16 18 20 


i = o.S i.o 1.2 1.4 


16 1.8 2.0 2.2 


2.4 2.6 2.8 


d = 24 28 32 36 40 J48 56 J64 {72 


80 


-f =•- 3-2 3 6 4-044 4.8 5-2 5-6 6.0 6.4 
1 11 1 1 1 1 


6.8 



iI5G7S 1012141G1S20 24 28 32 3G <10 -JS 

Fig. 2o8. 



ao liO 61 



Diameter and pitch both in millimeters. 

For any interpolated diameter the next lesser 
72 i't, ordinate is to be taken, as for example d = 60. 




Pf.\i.z-Saarbruck System. Fig. 209. 



%(d-I)j.J 



d = 


6 


7 8 10 12 i4Ji6 


18 20 22 24 


s = 


1.0 


1.2 1.4 1.6 1.8 2.0 2.2 


2.4 2.6 2.8 3.0 


d = 


26 
3-2 


28 32 36 40 48 56 J64 J72 80 


s = 


3 6 4.0 4.4 4.8 5.4 6.0 6.6 7.2 7.8 



No interpolation to be made. 



C7S 101211101S20232a2G25 32 36 40 

Fig. 209. 




DelisIvE //. Fig. 210. 



d = 


6 


8 jio 


12 14 16 18 


20 24 


s = 


1.0 


1.2,1.4 


1.6 1.8 2.0 2.2:2.4' 2.8 


d = 


28 32 36 


40 48 56 64 


72 So 


s = 


3.2 3.6 4.0 


4.44-85.25.6 


6.o'6.4 



« 6 8 1012U1G1820 24 2S 3i 30 40 

Fig. 2IO. 



Ot) lil) Gl 



In all three systems the superficial pressure is quite satisfac- 
~tor3f. According to formula (74), taking 6' = 3600 we obtain for 
lvalues of/ — 

Delisle I S600 to 11,500 lbs. 

Pfalz-Saarbriick . . 8600 to 11,000 " 
Delisle II ... . 7600 to 10,000 " 

?79- 
New Systems. 

A thorough investigation of the proposed systems of the Ger- 
inati Society of Engineers failed to produce any definite results, 
and the whole subject of a metrical screw thread system is still 
unsettled. For this reason it has been thought advisable to of- 
fer a further discussion of the problem.- 

It might seem a shorter plan to adopt some one of the three 
preceding systems, yet they all seem capable of improvement. 



*This is especially necessary for use in technical instruction, which will 
■afford the surest method of introducing a metric screw thread system into 
practical use. The advocates of the Whitworth sj-stem urge the desirability 
■of an international standard, in view of the widespread use of the 
American system, which is indeed already in use to some extent in Ger- 
many. In this case the conflict between the two systems of measurement 
"has been met by proposing to take for any dimension in English units the 
next higher dimension in millimetres. Such a system would be impractic- 
able for educational purposes and would lead to many errors in actual 
practice. It also seems only to be practicable for the old Whitworth scale, 
and for the new scale, with its close divisions, its application would be im- 
possible. A comparison between the preceding diagrams will show that a 
close adherence to the Whitworth system would result in a complication of 
•dimensions -which would be most undesirable. 



For any interpolated meter the next greater 
ordinate is to be taken, as for example d = do.* 



The subject will bear further investigation in two main points 
one being the gradation of diameters and the other the profile 
of thread. The actual diameters and their gradation are of 
more practical importance than the gradation of threads. This 
is shown by the fact that the Whitworth profile has long been 
in use with the bolt diameters taken in Prussian inches, and 
more recently with dimensions in millimetres with Whitworth 
profile. One of the first requisites of such a series is that the 
diameters should follow the decimal divisions (see the third 
condition of § 77). This point is not met by the preceding sys- 
tems, since they lack the natural divisions 30, 50, 60 and 70. 
The removal of this objection introduces a new difficulty, but 
not an insuperable one. 

The critical feature of the screw thread system is really the 
relation which the diameter bears to the profile. A thread should 
not be said to be coarse or fiue, implying the ratio s : d, but 
rather should the depth of thread be considered, or the rati oi : d. 

This can best be illustrated by an example : 

If we select two equal sizes from the systems Delisle I and 1 1, 
we shall find that for the same pitch the threads are not alike. 

l{ d^ 60 mm. we shall have (see the dotted lines in Fig. 208 
and 210) in both cases 5=: 5.6, hence the angle of thread is the 
same. 

The working depth /, however, is : 

in 1 : /^^io ^0.65 .5 = 3.64 mm. 
in II : if ^ 3^/0 ^ 0.75 i = 4.20 mm. 



■^In both his systems Delisle has provided for the interpolation of inter- 
mediate diameters, but these have been omitted from the diagrams and 
tables to avoid obscurity. 



54 



THE CONSTRUCTOR. 



This gives for the diameter of the bolt at the bottom of the 
thread 

in I : d^= 57.70 — cross section 21S2 sq. mm. 
in II : fl'i= 51.60— " " 2091 " " 

■which shows a difference in resistance of about 5 per cent, be- 
tween the two bolts, the second having the coarser thread. We 
see here that a choice of the relation of jt to d affects the pro- 
file of thread, and it is this which led Delisle to suggest two 
systems. 

Whether the angle of 53° S' is preferable to the Sellers angle 
of 60° is uncertain. Among the preceding systems may be 
noted two for the latter, five for the former, and three for 

^ d 




still smaller angles ; and if the choice be given, it seems rather 
better to go below the Whitworth angle of 55° than above it. 
We prefer the angle as shown in Fig. 211 : 
2 /3=S3° 8' or to =s \ 



(77) 

(7S) 
(79) 



and hence 

For sizes of d from 4 to 40 mm. the pitch may be 
.s^o.4 + o.i rf .... 

and for sizes of d, from 40 to So mm. and over* 

s=2 -\- 0.06 d .... 

with the following series of diameters : 
4 5 6 7 S 
12 14 16 iS 20 22 24 
36 40 . 45 . 50 

Formula (78) is the same as in Delisle II, from 6 to 40 mm. 
Interpolation for intermediate diameters seems unnecessary ; 



,.(--tn-*i 



9 


10 


26 


28 30 


60 


70 80 




' ^ V 


. + 






.^ 


l<t >l 


A 






.*t> 


1 1^- — 


-*f -> 


< 


■""""*'" ~ 


V 


t< 




Cl=.30... 




; 



Fig. 212. 

should it be done, however, the formula should not be departed 
from, since the values in the second and third groups above 
■will give round numbers, and offer no difiiculty for their pro- 
duction on the screw cutting lathe. If it is still desired to use 
the angle of 60°, and yet retain the other proportions, we may 
take 

for rf^4 to 8, 5' =0.2 d (as in Delisle I) 1 

forrf ^ 8 to 40, s' ^0.8 \- o.\ d (as in Delisle I I . . (So) 

for a' = 4o to 80, .r'^ 1.5 4- o-oS d J 

in which arrangements the sizes 30, 45, 50, 60, 70 remain in the 
series, which may also be extended above 80 mm. The two 
plans may be compared to Fig. 212, in which the formulae are re- 
spectively applied to a diameter of 80 mm. The radii to the 



bottom of thread r\ and i\, are almost identical, as are also 
working depths, although the profiles differ, as shown by the 
triangles ABC and D E F. Instead of numbering the sizes 
arbitrarily, it seems preferable to use the bolt diameter for the 
number. Screw No. 20 would then stand for rf = 20 mm., No. 
4 for rf ^4 mm. Any establishment could omit numbers not 
desired without impairing the system, while for fine work_ 
smaller numbers could readily be added. 

?So. 

Nuts, Washers and Bolt Heads. 

The thickness of metal in a nut bears a close relation both to 
the depth of thread /, and to the pitch j. It is desirable that 
the formula to be used should give the dimensions readily ia 
order to avoid the necessity of approximating. 




Fig. 213. 

For the diameter D, of the inscribed circle of the hexagon we 
may take for finished nuts : 

Z> = .o4-l- rf-f 0.5.? ...... (Si) 

The maximum pressure upon the base of the nut in this case 
(for d = 3'') = about 2400 lbs. per square inch. Unfinished 
nuts are made somewhat heavier, and lor them we have 



A = o.i4"+'''+5-J 




J ..» 




Fig. 214. 



Fig. 215. 



Fig. 216. 



The use of the washer insures a better bearing for the nut in 
case the surface is not true. Its dimensions may be taken as 



diameter = U=d -\- 10s 

thickness ^ u = s 
4 



(83) 



*For sizes over So mm. -we have not yet established relations. If we take 
d= 150 mm. which is about as high as Whitworth or Sellers have gone, s^^ 
II mm., which seems a good proportion. See § 87. 



Bolt heads are often made square, but are preferable hexagon- 
al, and for them we may take D and Z)„ the same as for nuts^ 
and the height /« = 0.7 fl^. Fig. 213. 

For finished nuts the upper surface may be finished with a 
bevel of a frustum of a cone whose base =Z?, and a base angle 
of 30°, Fig. 214, or as a portion of a sphere with a radius of f 
D, Fig. 215, while unfinished nuts have the corners beveled off 
above and below, as shown in Fig. 216. 



THE CONSTRUCTOR. 



SS 



0,064 



0/Mrt 




Fig. 217. 
Bolts and Nuts. (Metric system). 



Bolt 
Dia. 


Pitch. 


Depth of 
Thread. 


Bottom 

Dia. of 

Bolt. 

dx 


Nut. 


Washer. 


Bolt 
Head 


Load. 
p 


mm. 


s 


t 


D 


A 


U 


u 


n 


kilos. 


4 


0.8 


0.60 


2.80 


9 





12 


I 


3 


16 


5 


0.9 


0.68 


3.65 


10.5 


— 


14 


I 


3 5 


27 


6 


I.O 


075 


450 


12 


— 


16 


I 


4 


41 


7 


I.I 


083 


5-35 


13-5 


— 


is: 15 


5 


57 


8 


1.2 


0.90 


6.20 


15 


— 


20 


1-5 


6 


77 


9 


1-3 


0.98 


7-05 


16.5 


— 


22 


1-5 


6 


99 


10 


1.4 


1.05 


7.90 


18 


21 


24 


1-5 


7 


125 


12 


1.6 


1.20 


9.60 


21 


24 


28 


2 


S 


184 


14 


1.8 


1-35 


11.30 


24 


27 


32 


2 


10 


255 


16 


2.0 


1.50 


13.00 


27 


,30 


36 


2 


II 


Z7,'i 


18 


2.2 


1.65 


14.70 


.30 


33 


40 


3 


13 


432 


20 


2.4 


1.80 


16.40 


33 


36 


44 


3 


14 


.53S 


22 


2.6 


1-95 


18.10 


36 


39 


48 


3 


IS 


655 


24 


2.8 


2.10 


19.80 


39 


42 


52 


3 


17 


784 


26 


3-0 


2.25 


21.50 


42 


45 


56 


4 


18 


841 


28 


3-2 


2.40 


23.20 


45 


48 


60 


4 


20 


1076 


,S0 


3.4 


2.55 


24.90 


48 


51 


64 


4 


21 


1240 


32 


3.6 


2.70 


26.60 


51 


54 


68 


4 


22 


1415 


36 


4.0 


3.00 


30.00 


57 


60 


76 


5 


25 


1800 


40 


4.4 


3-30 


3340 


63 


66 


84 


5 


28 


2231 


45 


4-7 


3-53 


37-95 


70 


73 


92 


6 


32 


28S0 


50 


5-0 


3-75 


42.50 


76 


79 


I'X) 


6 


35 


3613 


60 


,S.6 


4.20 


51.60 


89 


92 


n6 


7 


42 


5325 


70 


6.2 


465 


60.70 


102 


105 


132 


7 


49 


7369 


80 


6.8 


5.10 


69.80 


"5 


118 


148 


8 


56 


9744 



J Si. 

Table and Proportional Scale for 
Metrical Bolts and Nuts.-' 

The preceding table contains a summary of 
the preceding discussion, aud Fig. 217 is a dia- 
gram in which the relations of the parts are 
shown graphically. The value for j- is shown on 
a five-fold scale. The dotted line gives the value 
for s' of formula (80) . 

The diagram Fig. 217 shows the pitch of 
thread and the pressure upon a unit of area, for 
the dimensions of nuts and bolt heads for the 
preceding metric screw thread system for diam- 
eters from 4 to So mm. 

The pitch is shown five times full scale (line 
E) and ten times full scale (line F)\ the bolt 
diameter in its actual size (Hue/^), all measured 
from the base line A. The line i? is i mm. 
from D, and Cis 4 mm. from D while the dis- 
tance between A and G is 0.7 d. 

The various details may be summed up as 
follows : 

Between A and E = the fivefold pitch, 
" E and B = dia. of finished nut, 
" E and C ^ dia. of rough nut, 
" i^and D = dia. of washer, 
" A and G = height of bolt head. 

The tangent of the pitch angle ranges from 
0.064 to 0.047, and the pressure per sq. mm. on 
the thread, from 0.46 to 0.67 kilogrammes. 

?82. 

Weight of Round Iron. 

The weights in the following table are given 
by the Formula 

G = 2.6i'j d', 

the bars being one foot long and d = diameter 
in inches. For cast iron, multiply the values 
in the taole by o 93 and for bronze by 1.092. A 
hexagonal rod whose inscribed diameter = d is 
1. 103 time the weight of a bar of wrought iron 
of the same diameter. 

Weight of Wrought Iron Rods. One Foot Long. 









I 



.■63 

• 255 
.368 
.500 

•654 
.826 

1.02 

1.23 

1-47 

1.72 

2.0 

2.29 

2.61 

2.94 

3.31 



i>4 

lA 

'^ 
iH 

lit 

IlTT 
2 



3-68 
4.09 
4-50 
4-94 
536 
5.89 
6.39 
6.91 

7-43 
8.01 

8.57 
9.20 

9-79 
10.47 
11.02 



G 



2^ 
2tV 
2^ 

2tV 

2H 

2M 

ol 3 

2tt 

3 



11.82 
12.50 
13-25 
13-95 
14.76 

15-54 
16.36 
17.14 
1S.03 
19. 1 1 
19.79 
2061 
21-63 
22.52 
23-56 



?S3. 

Special Forms op Bolts. 

Instead of being made with square or hexagon heads, bolts 
are sometimes fitted with special heads, instances of which are 
shown in Figs. 21S to 222 ; the last being countersunk. These 
are all furnished with means to prevent the bolt from turning 
when the nut is operated. 



*This table has been kept in the metric system for obvious rea.=:ons. Trans. 



56 



THE CONSTRUCTOR. 



In Fig. ?23 is shown an anchor bolt witli cast iron plate for 
brickwork the bolt being inserted from above and locked by 
being turneO. 90°. The area of the anchor plate should in no 
case be ) e jS 'ian 100 d{- 

In Fitr 224 is shown a form of anchor bolt for masonrj' with 
a cast ;:".~"i washer, secured by a key. The washer should be 
not less than 25 d^ in area. Such plates are often made of 
wrought iron. 




Fig. 21S. Fig. 219. Fig. 220. Fig. 221. Fig. 222. 

In Figs. 225 and 226 are shown bolts secviredby cross ke3'S and 
side keys. In these two figures the nuts are shown iu different 
positions, the latter being the more convenient to use the pro- 
portions shown in Figs. 214 to 216. Figs. 227 and 22S are forms 




C",6a 







-ZJ^- 



Fig. 223. 



^vB^ 



Fig. 224. 



of stud bolts. Fig. 229 is a cap screw. For small work these 
cap screws are often made with cylindrical heads with slots for 
use with a screw-driver. 




?S4. 

Wrenches. 

A wrench is a lever adapted to tighten and loosen nuts and 
bolt heads. The simple wrench, shown in Fig. 230, in two 
forms, consists of a flat or round handle fitted to the shape of 
the nut, the dimensions being based upon the unit D, which is 
the diameter of the nut as given in formula (Si). The double 
wrench, Fig. 231, is arranged to receive nuts of different sizes at 
the opposite ends of the handle. If the ends are inclined so as 




Fig. 230. 



to bring the corners of the hexagon at 15° and 45° with the 
axis of the handle the wrench will be able to operate in con- 
tracted spaces by j'j revolution of the nut.* 




Fig. 231. 

§85. 

Not Locks. 

For bolts made according to the preceding proportions, the 
angles of pitch are not steep enough to allow the pressure in 
the direction of the axis of the bolt to overcome the resistance 
of friction and cause backward rotation. If, however, there is 
much vibration, lost motion niaj- appear and gradually cause 
the connection to work loose. This is true of foundation bolts 
as well as of those in moving parts of machinery and in loco- 
motive and marine engines. For these and similar cases it is 
necessary to have some method of securing the bolt or nut 
from coming loose, and a variety of such nut locks are here 
shown. 





Fig. 232. 



Fig. 233. 



Fig. 234. 



One of the oldest and most useful forms is the jam nut. Fig. 
232. Both nuts should be truly faced so that they will bear 
fairly upon each other. The thin nut is frequently placed 
under the thicker one, but this is immaterial since a nut of a 
thickness of 0.45 to 0.4^ is as strong as the bolt thread. The 
security obtained b}' the use of the jam nut is not ver}' great, 
and the form with right and left hand thread, as shown in Fig. 
244, is to be preferred when greater securitj' is essential. In 
Fig. 233 is shown a split pin, often used in connection with a 
jam nut. Fig. 234 shows an arrangement with a key upon the 
nut, making a very convenient and secvire combination. In 
the three preceding cases the action is such as to tighten the 
nut upon the thread. The three following methods are made 
to hold b}' fastening the nut or bolt, or both, to the parts which 
they are intended to hold together. Fig. 235 is used in the 
spring hangers of Borsig's locomotives. Fig. 2j6 on an oil cup 
lid, and Fig. 237 on a set screw for a connecting rod end, 
arranged to lock at any 1-12 part of a turn. 

In the following methods the nut is held from turning by be- 
ing locked to one of the stationary pieces, the bolt itself being 
secured in a similar manner. The form shown in Fig. 23S is 
used for bearing cap bolts, the support at the middle of the 



* This idea is due to Proell. 



THE CONSTRUCTOR. 



57 



split pin keeping it from bending. The method shown in Fig. 

239 is used for the bolts in a steam piston, while that in Fig. 

240 is for a bearing cap. The latter form is arranged by means 
of the sever notches, to lock at every y'y of a turn, while the 
other two require Y(, of a turn between successive positions. 




ElG. 235. 



Fig. 236. 



Fig. 237. 



Fig. 241 shows a device for securing the nuts of stufBng box 
bolts as applied to locomotive engines. The ratchet wheels are 
attached to the nuts,. and similar notched nuts may be used to 
advantage in many places. 




Fig. 238. 




Fig. 239, 



A method of securing the bolts for locomotive springs, used 
by Borsig, is shown in Fig. 242. The tension of the spring 
keeps the bolt from turning, and the cap which secures the nut 
is fitted to the end of the bolt as shown ; this locks for every Yf, 
of a turn. Fig. 243, shows a nut arranged to be locked by a 
set screw. This method, used by Penn, is a very useful form 




Fig. 241. 

for bearings, spring hangers, and other situations, since it per- 
mits any fraction of a turn to be made. The nut, in this case, 
should be a little thicker than usual in order that the lower 
cylindrical portion may not be too weak. The diameter Z),, is 
in this case taken from formula (82). The small set screw 
should be made of steel and hardened. This form of nut lock 
is especially useful on marine engines. 




Fig. 242. 



Fig. 243. 



A different class of nut locks depends for its action upon the 
introduction of an elastic resistance between the bolt and the 
nut.* The elastic washer of Pagel and similar devices have 
found many applications. Parsons' bolts belong to this class.f 



* See l^udewig- Nut Locking Devices. Bavarian Industrial and Technical 
Journal, iSyo, pp. 17, 144, 283 ; also Journal of the Society of German 
Engineers. 

t Engineer, July, 1S67, p. 16 ; Nov., p. 391 ; Engineering, 1S67, Nov., p. 411 ; 
Railroad Journal, 1S6S, pp. 77, 117. 



In this form the body of the bolt is fluted, so that the cross 
section is reduced to about the same area as that of the bolt at 
the base of the thread. This increases the elasticity of the bolt 
and enables the nut to be tightened so that it is much less likely 
to come loose. Fig. 244 shows a modification of this form 
used by Gerber for bridge connections. The security is still 
further increased by the use of a left hand jam nut. Instead of 
being fluted, the body of the bolt may be flattened on four sides, 
or the reduction of area may be obtained by drilling a hole 
into the bolt from the head to the beginning of the thread. 





Fig. 244. 



Fig. 245. 



One of the most important instances of screw fastenings may 
be found in the construction of built-up screw propellers, in 
which the blades of the scre%v are bolted fast to the hub, a con- 
nection requiring the greatest strength and security. Fig. 245 
shows the base of such a propeller blade, from the same 
example as shown in Fig. 192. The flange of the blade issecur- 




FiG. 246. 

ed to the hub by sixteen cap-screws. Four set screws serve to 
provide a small adjustment of the blade within the range of 
motion of the oval bolt holes. 

All of the cap screws are secured. Fig. 246 shows the method 
adopted by Penn. The bolts, which in the case of the Minotaur 
are 3J4" diameter, have a common ring washer under the 
heads. When the bolts have been screwed up as tightly as pos- 
sible, a ratchet washer with hexagonal hole is slipped over 
each bolt head. These ratchet washers are prevented from 
turning by the introduction of small locking pieces which are 
bolted fast to the large ring washer, being held down by the 
thin nuts shown. The ratchet washers have 11 teeth, and 
hence each bolt may be locked at -^^ P^rt of a turn. Fig. 247 
shows a method by Maudslay. Here each pair of bolts is held 
by a flat key which permits looking at j'j part of a revolution. 



58 



THE CONSTRUCTOR. 



A continuous washer ring is not used with this method, but one 
washer is put under each pair of bolt heads, to which the lock- 
ing key is bolted. Another method by Maudslay is shown in 
Fig. 248. A double washer is placed under two adjacent bolt 



Fig. 247 




Fig. 24S. 



heads, and each bolt is locked by a small block held against 
one of the faces of the bolt head by a small bolt. Three bolt 
holes situated 40° apart are tapped in the washer for each block, 
thus giving an adjustment of xV of 3. turn. The method by 
Penn gives the best opportunity for adjustment. 



Speciai, Forms of Screw Threads. 
Screw threads of siiuare or trapezoidal section may be used 
for bolts, but in their use it is desirable to use a deeper nut in 
order to secure a sufficient number of threads in the nut to keep 
the pressure per square inch on the thread within the pre- 
scribed limits. Trapezoidal threads are well suited for bolts, 
since the relation between 5 and d permits the use of the same 
proportions as those given for V threads in Fig. 211. In fact 
the thread in Fig. 250 may be given the same proportions as 
that in Fig. 211, for depth t, and pitch i, making the angles 
respectively equal to 0° on one side and 45° on the other. 
These forms of screw-threads are principally used for screw- 
presses and for similar uses. 




Fig. 24.9. 
For such screws the diameter (/,, at 



Fig. 250. 

the bottom of thread, 
If, however. 



is generally determined from formula (72). If, however, it is 
desired to make the diameter d^ a niimimum, we must consider 
the pressure to act only on one side of the thread in the nut 
and then take the pressure per square inch at double the previ- 
ous allowance, or 1 = 7110 lbs. We then have, 

rf, = 0.0134 v''/^) ,_ , 

Passes,/; I («^> 

The depth of thread, both for square and trapezoidal threads, is. 






and for square threads- 






'L 
5 

and for trapezoidal threads- 



8 
4 



— ^ i— '^' 



(85) 



Formula (84) is applicable to screws of locomotive springs, 
since in this case the conditions are well complied with. 



In order that the nut may not wear or grind out, the working 

pressure on the threads should not exceed say 700 lbs. per square 
inch. These conditions will obtain, according to (73), when the 
number of threads //, in the cast iron or bronze nut is not less 
than 



If / = 



= 0.0014 



-d, we have 



0.0003554(1-34) 



n = 0.00245 -5'= 0.00312 



dl 



(86) 



(Sy) 



The depth of thread, from (S5) = 
rf = 3.9i2", or about 3- 



= 0.392", which gives 



Example. For a pressure of 55.000 lbs., we have, under the preceding for- 
mulEe, from (84) the diameter at the bottom of the thread 

«',= 0.0134 v^/'= 0.0134 X 234.5 = 3.14" 

S 

il// 
16 ■ 

From (87) we have, makings ^ 7710 lbs , themimimum num- 
ber of threads in the nut n = 00245 ■5'= 17.4 which gives for the 
height of the nut for square thread h = ns = I7.4X.7S5^ 13.65", 
while for trapezoidal thread /; = 17.4 X .523 = 9. i". 

In many cases the diameter of such screws is made greater 
than the normal diameter indicated in the preceding discussion 

! i 



r 





Fig. 251. 



Fig. 252 



for the given load. Such screws may be called enlarged.screws, 
as compared with the normal dimensions as previously deter- 
mined. For such screws the same cross section of thread and 
the same height of nut may be given as for the normal screw 
of the same load, in which case the wear will practically be the 
same for both examples. Enlarged screws are frequently used 
for presses, where the diameter must be made greater than indi- 
cated by formula (84) in order to resist bending stresses. 

I 87. , 
Screw Connections, Fi.ange Joints. 
In screwed connections a distinction may be made as to 
whether the force acts parallel to the direction of the axis, or 
at right angles to it. The latter condition, which produces 
shearing stresses, is shown in the examples given in Figs. 251, 
252 and 253. If we take d, as the diameter of the rod through 
which the force acts, we may call d', the bolt diameter, and 




253. Fig. 254. 

then determine their relation for various cases. In Fig. 251, 
d'^d ; in Fig. 252, d'^\.\ d ; in Fig. 253, d'^^=d ; the increased 
diameter for Fig. 252, being given because it is possible in that 
case for the load to act so unequally that the greater portion 
may pass through one of the rods. Fig 254 shows a tumbuckle 



THE CONSTRUCTOR. 



59 



■with right and left hand thread. In this it is desirable to make 
the nut somewhat deeper than d, as shown. A form of junc- 
tion piece for a point where four members meet is shown in 
I'ig- 255. Such examples as the preceding are of frequent oc- 
currence in bridge and roof construction.* 




Fig. 255. 



Fig. 257. 



Bolt connections which bring shearing stresses upon the 
bolts are of frequent occurrence in bridges built with pin-con- 
nections, the general method in use in America. These designs 
exhibit very fully the substitution of bolt or pin-connections 
for riveting, and the method has been carried to great perfection. 
Some examples are here given. Figs. 256 and 257 show an in- 
tersection of several members of the bridge over the Ohio, at 
Cincinnati. The top chord and the posts are double, and are 




tys i lyi 



Fig. 258. 



made of plate T and angle iron. The diagonal rods and braces 
to resist the action of the wind are connected to the upper 
chord by means of a bolt passing entirely through the beams 
and threaded at both ends. The nut on the left end is in the 
form of a fork to receive the ends of the braces, while the right 
hand end is fitted with a thin octagonal nut. The ends of the 
braces are held by a bolt passing through the fork, with a nut 
at each end. The pins are carefully turned and closely fitted ;t 
after years of service they show no signs of looseness. J The 
proportions are such that stress on the bolts does not exceed 



* other good exaiuples of similar work in roof construction may be found 
in E. Brandt's "Iron Constructions," Berlin, Ernst and Korn, 1S71, 2d 
Edition. 

t It is well known that variations in temperature during the borinar of the 
holes for the pins in the eye bars may make suiificient difference to mater- 
ially affect the fit. This has been overcome by the use of a double boring 
machine which the author saw at work in the notable bridg-e works at 
Phoenixville. whereby both ends are bored simultaneously, the distance 
being gauged by a wrought iron jig bar, which varied in length to the same 
extent as the eye-hars themselves. 

X See H. Fontaire. " llndustrie des Etats Unis," Paris.Baudrj', 187S. Rol- 
ler, Highway Bridge's New York, Vv'iley, 1S7S. 



15,000. lbs, in most cases not more than 10,000 to 12,000 lbs. The 
e connection of the posts to the chords (in the illustration the riv- 
ets are omitted) is both simple and strong. The posts are provided 
with cast iron ends, which are fitted with square projections en- 
tering into the tops of the posts ; in these capitals are wrought 
iron dowel pins which pass through the lower angle iron and 
lower plates of the top chord. The diameter d, of the main 
bolts varies from 4 to 5;^ or 6 inches or even heavier, according 
to the load. Their dimensions are based upon as bearing stress 
of 8000 lbs., while the diagonal braces and the lower chord are 
proportioned upon a tensile stress of 10,000 lbs. (a ratio of o.S, 
see I 5). The compressive stress in the top chord is about 8,500 
lbs., and in the posts, owing to the bending action, only about 
5000 lbs. 

Fig. 25S shows an intersection on the lower chord of the 
Niagara railway bridge (9 spans over a total width of stream of 
about 1900 feet). In this case the posts and top chord are 
made of the ingenious Phcenix column of quadrant iron. The 
illustration especially shows the method by which the cross 
beams are connected to the longitudinal members. In this 
case the stress in the body of the screw bolts is about 8000 lbs., 
rather more than given for press screws in \ 86. A cast iron 
base, through which the large pin bolt passes receives the 
thrust of the post, and to it the cross beams of I shape are 
bolted. On these cross beams are wooden stringers to which 
the roadway is secured. 

It will be noticed that these examples of bolt work far ex- 
ceed the limit of size set by the Society of German Engineers 
for bolt dimensions, viz., 80 mm. or 3fg". Should such 
sizes be necessary the formulae in \ 79 should be reconsidered. 
Pig. 259. Fig 260. F7G. 261. 





Fig. 262. Fig, 263. Fig. 264. 

In uniting the various parts of iron constructions, flange 
joints are very frequently used. These are made in a great 
variety of forms for various conditions. The following figures 
show some examples of comer junctions with flanges. Fig. 
259 shows three external flanges, with a dished base. Fig. 260, 
also three external flanges, with an external plinth on the base. 
Fig. 261 shows one external flange, and two which are half 
external and half internal. Fig. 262 has three half external 
flanges and a base as in Fig. 260. Fig. 263 has also three half 
external flanges, and Fig. 264 two external and one half- 
external flange. The last three examples produce a more 
pleasing external appearance than the preceding forms. If the 
form shown in Fig, 262 is used, with the flanges all turned 
inward, the bolts cannot be unscrewed from without. 

Proportions for flange joints are shown in Fig. 265, the bolt 
diameter d, being obtained from the thickness of metal ^. The 
distance between bolts is usually lyi to 3 /), i? being the 
width of the nut across the flat dimension. The width of 
flange is given in the illustration for metric sizes = 10 mm. + 
2.8 & = Yi" + 2.Z6. _ 

If the flanges are finished on the planing machine, a ledge is 
left for finishing, as shown on the left of Fig. 265, in order that 
a' fair bearing may be secured. Flange joints which are to 
be bolted together without finishing are made as shown in Fig. 
266, with a gasket of some form of elastic packing. Such 
flanges are sometimes made for vessels with very thin walls, and 
on the left of Fig. 266 is shown the method of assembling a 
cylindrical vessel, such as a water tank. The base has internal 
flanges for the bottom pieces, with an external flange for the 
connection to the body. By turning the flanges of the bottom 
inward a flat exterior base is obtained, well adapted to sustain 
the load of the water. The walls are very light, (5 = only about 
yi" , the bolts are Y%" diameter, and their distance from centre 
to centre, in the base, 13.5 d. and in the vertical joints of the 
walls 15 d, and in the circumferential joints 20 d. 



6o 



IHE CONSTRUCTOR. 



i 88. 
Uni<oaded Bolt Connections. 

Various methods have been adopted to relieve bolts, in a 
connection, from the direct stresses due to the load, much in 
the same manner as has been described in g 71, for keyed con- 
nections. In Figs 267 and 26S are shown methods of notching 
two plates together. The bolts are relieved from the action of 
tensile or compressive stresses which act normal to the direction 
of the tongue and groove. 

Fig. 269 shows a method of constructing a prismatic intersec- 
tion so as to relieve the bolts from transverse stresses ; while 



through both plates, the diameter d at the thread being 6j/jf" , 
and in the body 5}i" . The shoe is tongued into the sole plate 




Fig. 265. 

Fig. 270 shows a very convenient and useful form in which the 
projections on each piece lip over the other, greatly increasing 
the security of the connection. The bar may be made of wrought 
iron and the fitting. should be made to conform carefully to the 
position of the bolt holes. 

If the parts are large they are often both made of cast iron, 
and in some cases a turned dowel is let into both parts. The 



0,6i, 





Fig. 266. 

constructions shown in Figs. 269, 270, are used in the frame- 
work of large water-wheels, in which case the lower piece is 
made flat, thickened wherever it may be fotmd necessary. 

In many cases the lateral stresses are not great, while at the 
same time it is not desired that the bolts shall be made to fit 
closely. In such positions dowel pins are frequently used, 
being made of steel and fitted to reamed holes. 

An example of bolt connection of large proportions, in which 
the lateral stresses are relieved, is shown in Fig. 271. 

This is taken from the bridge over the Mississippi at St. 
Louis, and shows the bearing of the end of the lower member 
of one of the arches, which are composed of steel tubes. There 
are four such bearings at the end of each arch, or 24 bearings in 
all. The shoe to which the end of the tube is fitted is made of 
■wrought iron, and the sole plate, of cast iron. Three bolts pass 




Fig. 269. 



Fig. 270. 



and the latter is supported by the masonry of the pier. The 
hole across the .shoe is for the reception of the bolt by which 
the adjoining bearings are braced together. 

..Mh , 




Fig. 271. 



CHAPTER V. 

WUXNALS. 

Various Kinds of Journals. 

? 89. 

Journals are made for the purpose of permitting parts of 

machinery to rotate about a geometrical axis and hence they are 

necessarily round, and their use involves some form of bearing 

or box for a support. 

A journal may be subjected to pressure upon its side, or rather, 
normal to its axis ; or the pressure may act lengthwise, in the 
direction of the axis. This gives us the two divisions : 

1, Lateral journals. 

2, End, or thrust journals. 

In the calculations relating to these, both the questions of 
strength and of friction must be considered. In machine con- 
struction many forms of journals are employed, the most 
important of which will be here considered. 



THE CONSTRUCTOR. 



6i 



a. lateral jo urn a ls. 

Overhung Journai^. 

? 90 

A lateral journal which is connected on one side only to the 
member to which it belongs is said to be overhung. Such jour- 
nals are usually made cylindrical, as in Fig. 272, with a collar 
at the outer end, the height of the shoulder e above the diame- 
ter d being — 

e= 14" -{- 0.07 (/ (88) 

If the lateral pressure = P, the length of the journal ^ /, 




K..;e i 



ElG. 272. 



and the permissible stress at the root of the journal = S, we 
have for the diameter to resist the pressure 



'=/4(l)/T 



(89) 



The ratio of / : </, determines the superficial pressure between 
journal and bearing. In ordinary circumstances the pressure per 

p 
unit of area on the lower half of the bearing is/ = j-r . When 

the journal is revolving, this pressure is not the same at all 

points, but has at the base line a value = p' =: — /, and at 

4 
any angle ft from the base line, a value/' = — /cos ft Since 

the relation between /" and / is constant, we may use the lat- 
ter value for all purposes of calculation. 

For any given value of /, we have from the preceding : 



d T 16 / T 






(90) 



In order that the wear may not be too great at high rotative 
velocities, it is advisable to take/, somewhat less than the max- 
imum value given above, and it may be made proportional to «, 
the number of revolutions per minute, or : 



= {l 



(9') 



in which a is a constant, dependent upbn material and lubrica- 
tion. By combining (91) with (89) we get: 



'-f^T/ 



P 



(92) 



These four equations should be applied and the greater values 



of d, and — ;- used. The maximum value of * = . 

' d '^ a 

For the value of the constant, the following considerations 
obtain. If the pressure on the journal acts constantly in the 
same direction it produces a higher superficial pressure on the 
lubricant than when, for example, the pressure is reversed 
frequently, as in a steam-engine crank pin. In the latter case 
there exists a kind of pumping action between the journal and 
bearing, which constantly draws the oil into the bearing surfaces, 
keeping them lubricated so that a higher value of / may be 
taken than when the pressure is acting continuously in one 
direction. Such bearings, however, are frequently subjected to 
violent thrusts and shocks, so that a lower value of 5 should be 
taken than with journals in which the directions of pressure is 
constant. For journals which only make a partial revolution, 
much higher pressure may be permitted, than for revolving 
journals. The former may be classed as journals at rest, as dis- 
tinguished from running journals. The constant a in equation 
(91) must be determined from practical considerations. It will 
be found that in practice, wide variations in the value of a 



occur, but while fair results are obtained with the smaller values, 
the increased value of a secures greater durability. Good 
lubrication is of the highest importance, and especially a good 
distribution of the lubricant over the bearing surfaces. 

For bronze bearings under favorable conditions when the 
pressure is constantly in one direction, a may be taken = 75, 
while if the direction of pressure is periodically reversed, a may 
be taken = 150. 

The following table will give the general proportions for 
lateral journals : 



sAII 



gAII 



PROPORTIONS OF JOURNAI.S. 
Wrought Iron 



Constant Pressure. 
Cast Iron. 



po = 8500 

5 = 8500 

o.oi7v'i» 



d 

d = 



S = 

_/_ _ 

d ~ 

d = 



"rt 




/ 


= 













10 


i' 







f-' 




to 


Vil^ 


/ 




c 




d 




3 

Si 




d 


= 



Wrought Iron. 
pa = S500 

S5OO 
0-5 

o.oi7v/p 



Wrouglit Iron. 

700 

S50O 

1-5 

O.OS'j/P 



4260 
4260 

o.o24Sv^ 

Jtiterjnittent Pressure. 
Cast Iron. 

4260 

4260 

0.5 

0.0245^P 

.onstant Pressure 
Cast Iron. 

360 
4260 

o.o43v/p 



Intermittent Pressur 
Wrought Iron. Cast Iron. 



r 
p = 

5 = 

M VI1 1 _L ^ 

I M rf = 



1422 
7000 

I.O 
0.027.^J. 



700 

3500 

1.0 
0-03 7v/p 

Constant Pressure. 



s = 

J_ _ 

d " 

d = 



Wrought Iron. 

75 
8500 

O. I3V'n 



Sted. 

14,220 

14,220 

0-5 

°-oi35v/p 

Stce!. 
14,220 
14,220 
0-5 

o-oi35\/p 

Steel. 

700 
14,220 
1.94 
0.027.y/p 

Steel. 

1422 
11,840 
1-3 
o.024v'> 

Stee. 

75 
14,220 
0.17^/^ 



0.0244 



/? 



Vp 



0.019 



/? 



x/> 



Interinitteni Pressure. 



s = 

/ _ 

d '~ 

d = 



0.0273 



Wrought Iron. 
150 
7000 

o.o8v'i 



Steel. 

150 
11,840 

o.iov';; 



sf' 



Vp 



0.02 Y/4^/? 



If n > 150, the ratio of I : d, is first approximatad and the 
value substituted in the last formulas of the table. 



62 



THE CONSTRUCTOR. 



For hollow journals the following proportions may be adopted. 
Let (/(, = the external and d-^ the internal diameter of the 



equivalent solid journal, i/' 



d 



we have : 



* 



(94 



the length of both solid and hollow journals being the same. 




If, however, the ratio of diameter to length is to be the same 
then 

d„ ^ I 

d 



f 



i> 



(95) 



from which the following series is obtained. 

rfi : rf(, = V ^ 0.4 0.5 0.6 0.7 0.75 0.8 



^ I 
:v/r 



-t 



I* = I.OI 



1.05 
1.06 



1. 10 1. 14 1. 19 
I.21 i.;,o 



.■^i = I.OI 1.03 I.Ob 1. 15 

In both cases there exists a smaller superficial pressure for 
the hollow journal than for the solid one. A common ratio of 
internal to external diameter is 0.6, and such journals were fre- 
quently used in cast iron work and are again being used in con- 
nection with hollow steel shafting and axles. 

Bronze boxes or their substitutes, such as white metal or 
other combinations, belong more especially to the subject of 
bearings (§ 96), and their use permits a higher superficial pres- 
sure without creating an excessive increase in the coefficient of 
friction. For moderate speeds, boxes of cast iron give results 
which are as satisfactory as can be obtained with bronze. This 
is especially the case with machines which are actuated by hand. 
For heavier or continuous service cast iron boxes are only suit- 
able when the pressure is not great, and examples of such 
bearings will be given in a later chapter. Bearings of wood 
may be operated satisfactorily at a pressure double that which is 
used with bronze, if the journal runs in water, or is kept wet. 
For heavy mill shafting making from 60 to So revolutions per 
minute, wooden bearings lubricated with grease are often used. 
For mill spindles, boxes with bearings of willow wood are 
sometimes used with good results. In this case the speed some- 
times exceeds 100 revolutions per minute, but the pressures are 
light. 

§ 91- 

ExAMPi^Es AND Tables of Journai^s. 

In the following tables are collected the results of the for- 
mula (93) in which the number of revolutions of the journal is 
assumed to be not greater than 150. 

1. Example, a water wheel weighing 66,000 pounds carries a load of 212 
cubic feet of water. The axis of the wheel is of cast iron, and the load is 
equally distributed between the two journals, giving -a load upon each 
journal of 33,000 4- 6605 = 39,605 lbs. The nearest value to this in the table 
is 40,05s lbs., which would give a diameter of .S^ inches, and a length of 
12^ inches. 

2. Example. A wrought iron shaft for a similar load, but subjected to 
alternating action, should have, according to the table, a diameter of about 
57-^", and the same length. 

If in cast steel, with alternate action, diameter should be about 4^ inches, 
and length of 4.75 X i-s = 6.175". 

3- Example.* The centres of the walking beam of the water engine at 
Eleyberg in Belgium each bear a load of 309.210 lbs. The journals are hol- 
low, with a ratio of external to internal diameter of 0.5. We have from 

(93) and (94) 

do = 1.02 X 0.043 v 309,210 ^ 24. 3S" 
and a length lo = 24.3S X 1.5 = 36.57" 

which gives a pressure of about 350 pounds per square inch of projected area. 

The actual dimensions of these journals are o'o^ig^^'", l^=^\%" , 
which gives a stress at the base of the journal of a little over 
4000 lbs., but the actual bearingis only \S}i" long, which gives 



a pressure of nearly 1000 lbs. per square inch, whicli appears to 
be too great ; and in actual practice these journals are obliged 
to be kept cool with water. 

In actual practice there is very little uniformity in the pro- 
portions of journals. Sometimes the distinction between con- 
stant and alternate action of load is considered but often it is 
neglected. In the case of locomotive crank pins, for example, 
p is frequently as high as 1500 to 3000 pounds per square inch, 
and on the cross head pin, as high as 4500 pounds. On the 
other hand quite low values of p are sometimes found on the 
crank pins of marine engines, f In all cases careful lubrication 
is of the utmost importance. When the number of revolutions 
is very great the length of the journal should be made greater 
than is given above. 

Table of Journals. 
Value ot P. 



1.25 
T..50 

1-75 
2.00 
2 25 
2.50 
■75 
3.00 
325 
3-So 
4.00 
425 
4.50 
4-75 
5-o> 
5-50 
6.00 
6.50 
7.00 

7-5° 
8.00 
8.50 
9.00 
9.50 
10 00 
10.50 
It. 00 
11.50 
12.00 



0.20 
o 20 
0.25 
o 25 
0.28 
0.28 
0.32 
0.32 
032 
0.36 
0.40 
0.40 
0.40 

044 

0.46 

048 

0.50 
0.52 
0.60 
0.62 
0.64 

0.68 

0.72 

0.74 
0.76 
0.80 

0.85 

o.go 
0.92 
0.95 



Direction of Load Constant. 



Wrt. Iron Castlro n 

/ _ I 

~d~^ 



Steel 



1121 

1752 

2523 

3434 

4485 

5677 

7009 

8481 

10093 

1184s 

13738 

17943 

20256 

22709 

25303 

28036 

33924 

40373 

4738' 

5495 

63082 

71773 
81025 
90838 
101212 
112141 
123641 
135696 
148313 
1 61 489 



554 
866 
1247 
1698 

22l8 

2807 

346s 

4193 

4989 

5856 

6792 

8870 

10014 

11227 

12509 

13860 

16771 

19959 

=3424 

27167 

31187 

34483 

40058 

44909 

50037 

55413 

61 1 26 

67087 

73324 

79838 



1419 
2217 

3193 
4346 

5677 
6870 
8871 
10734 
12774 
14992 
17387 
22709 

25637 
28742 
32025 
35484 
4=935 
51096 
59967 
69548 
79838 



102520 
11491S 
128097 
141935 



171741 
187709 
204386 



Direction of Load Varying. 


Wrt. Iron 


C'tlron 


Steel 


I 


I 


I 








d 


d 


d " 


1419 


724 


1833 


2217 


1113 


2i83 


3193 


1629 


4124 


4346 


2218 


5163 


5677 


2896 


7331 


6870 


3666 


9278 


8871 


4526 


11455 


10734 


5476 


13861 


12774 


6517 


16495 


14992 


7649 


19359 


17387 


S870 


22452 


22709 




29325 


25637 




33106 


28742 




37115 


32025 




41353 


35484 




45821 


4293s 




55443 


51096 




65982 


59967 




79260 


69548 




89809 


79838 




103097 


90868 




117301 


102520 




J32422 


114915 




148460 


128097 




165413 


141935 




183284 


156483 




202070 


171741 




221773 


187709 




242394 


204386 




263934 



4. Bxample. An axle on a railway carriag^e makes from 200 10300 revolu- 

/ 
tions per minute ; n may taken = 270, and from_(g3) we have —y 



o.i3\/ ; 



270 

^ 2.14. In practice the ratio is made from j.8 to 2.0. The journals of fan 
blowers are often operated at more than 1200 revolutions'; hence we get, in 

/ ,— . / ' . 

such cases — r- =0.13 \/i2oo ^4,5, or for steel ---^0.17 v 1200:= 5,5. The 

blowers made by Sturtevant, of Boston, have steel shafts, with the journals 
5 to 6 diameters in length. 

? 92. 

Neck Journals. 

When a journal is placed between two loaded parts of a shaft, 
as shown in Fig. 274, it is called a Neck Journal. 






f<.-l- 



FiG. 274. 

In such cases the diameter^-' is dependent upon other condi- 
tions than those of mere pressure. In order that the wear 



* Portfeuille de Johu Cockerill, I. p. 18 



t See Marks, " Crank Pins and Journals," Philadelphia, Kildare, 1878, 
where the following values of/ are given : Swatara, 400 ; Saco, 412 ; Wamp- 
anoag, 725 ; Wabash, 470. The third of these engines had a cylinder 100" 
diameter, and crank pin 16'' dia., 27" long, and the stress in the preceding 
cases was respectively, 4039, 3071, 10,537, ^nd 2745 lbs. 



THE CONSTRUCTOR. 



63 



may not be greater than in the case of overhung journals, the 
conditions of speed, lubrication, bearing metal, being the same, 
the length should not be made less than the corresponding 
end journal. If it is practicable to make the length greater, it 
may be done to advantage, and the weai thereby greatly re- 
duced* In many cases, however, the lack of space limits the 
length, as for example, in the case of crank axles for inside 
connected locomotives. Such journals are properly considered 
merely as enlarged end journals. 

For hollow journals of this type formula (94) may be used. 

I. Example. Borsig's Express Locomotive at the Vienna^ Exposition.! 
The journal of the rear drawing axle of steel was loaded with i3,2cx> lbs. 
d = iy%" , V = 7xV " According to the table the ^corresponding journal is 



Intermittent Pressure. 



■- 692.4 lbs. 



given &sd ^ 3H". ^= 3-^25 X 1-94 = 6.1", and_?> = ^:l = 

3.125 X 6.1 

In this case /' is much greater than /, and for the given values of/', and d* 
13,200 



■we have p = 



= 253.3 lbs. 



7125 X 7-3125 

while if /' = /. the pressure p — '3,2oo 



: 303 lbs. 



7.125 X 6.1 

2. Example. In the same locomotive the forward axle carried the crank 
pin journal upon which the entire force of the piston was exerted. The 
total pressure on the piston was 32,120 lbs., and the dimensions of the pin 
-were if -= ^Yz". I' = aVz". 

The corresponding values from the table of the preceding 
section give (/ ^ 4,',4 ",/ = 4, 25 x 1.3 ^S/^"/! ^ about 1400 lbs. 

The actual value oi p, for the sizes used is — '^-- = 1730 

4-125 X 4-5 
lbs. In this case /' is less than /, on account of lack of room, 
which accounts for the increase in superficial pressure. 

I 93- 
Fork Journai^. 

A Neck Journal which is held at both ends in a 3'oke or fork, 
as shown in Fig. 275, may be called a Fork Journal. Such 
journals may safely be made of lesser diameter than those which 
are overhung. If we let /'= the load, /^length, andrf = 
diameter, and s, the maximum permissible stress, we have from 
case VIII. I 6, 



^=/4/4 /' 



and if, as in the beginning of § 90, we put p =—p 



d 1 4P T / 



(96) 



(97) 



Proceeding as in J 90 we obtain the following collection of 
proportions. 

Formttlce for Fork Journals. 
Constant Pyessure. 
Wroughtjlron. Cast Iron. Steel. 

S500 4250 14,220 






& g 



fe o 



f Po 

s 
I 

~d 



po 

s 

I 



S500 



4250 
4250 



14,220 
I 



0.0121%//' 0.0171 -v//" 0.0095 v^ /" 



Intermittent Pressure. 
Wrought Iron. Cast Iron. 



8500 
8500 



4250 
4250 

I 



L d = 





- P 


= 


711 


V 


s 


= 


S500 


v.. 




I 

d 


= 


3 


s 










L d 


= 


0.0212 v//' 



o.oi2i\//* o.oi~}\y/ P 

Constant Pressure. 
Wrought Iron. Cast Iron 

355 
4250 



steel. 
14,220 

14,220 
I 

0.0095 \/ P 

Steel. 
711 

14,220 
4 



0.029 V/' 0.0185 \//' 



* See 1 109. 

fSee Berliner Verhandlung, 1874, p. 3S9. 







Wro'.ght Iron. 


Cast Iron. 


Steel. 


p 


= 


1422 




711 


1422 


s 


= 


7II0 




3550 


11,845 


I 
'd 


= 


2 




2 


3-5 


d 


— 


0.0185^/ 


P 


0.026 x//" 


0.0158 %/ 



High Speed journals of this sort are seldom used, and need 
not be considered liere. It wiU be noticed that these Fork 





Fig. 277. 



Fig. 275. 



Journals are comparatively small in diameter and of greater 
length ratio than the preceding forms. 

Bxample. A Fork Journal of wrought iron bears a load /*= 4400 lbs., act- 
ing constantly in one direction and revolves at a moderate speed. We have 

then if =0.0212 \/44co = 1.4") /=■ 1.4" >C 3^ 4.2". For an overhung journal 

under similar conditions we have, from the table of ? gi. d ^2", / = 3". The 
product of the length and diameter is approximately the same in both 
cases. If the length 4,2" is found inconveniently long, it may be diminished, 
providing d be proportionally increased. The strength will then be un- 
necessarily increased and the resistance of friction somewhat greater. 
These are only examples of the many variations which are to be met among 
the many conditions of practice. 

I 94- 

Multiple Journals. 

In some cases the resistance of friction becomes so great that 
a modification of the fork journal is resorted to in order to re- 
duce it within practical limits. Such an arrangement is shown 
in Fig. 276, which may be called a multiple journal. If we as- 
sume the load to be equally distributed among the plates, this 



-M 



Fig. 276. 

arrangement' may be considered as a series of fork journals. If the 
number of members on each side be taken = /\, each pair will 



support a A'th portion of the load P, and d will be 



/i 



times 



as large as would be required for a simple fork journal. 
If/<'= 2 3 4 5 6 7 8 



We have 



/J 



0.7 0.57 0.5 0.45 0.41 0.38 0.35 



Journals of this kind are generally of the slow-moving class, 
with a length ratio = i. The total length of journal is the^ 
2 A" d. Journals of this sort will be found is some varieties of 
chain links, of which examples will be giveu later. * 



* Joints of this kind may sometimes be subjected satisfactorily to a greater 
pressure than the calculation would indicate. Engineer VoUhering has 
used such a joint in a system of levers to operate a hea\'y drawbridge. In 
this case the load was about 95,000 lbs. IC= 10, the thickness of each plate |", 
d = 1^5, both plates and journal being of steel. 



64 



THE CONSTRUCTOR. 



I 95- 

Hai,f Journal. 

lu those cases in which the reduction of the moment of fric- 
tion is of great importance, the length of a journal may be 
somewhat increased, if the bearing surface is limited to one- 
half the circumference, as shown in Fig. 277, which shows such 
a bearing, the load acting constantly in one direction and the 
movement extending only through a small angle. In such 
cases it is desirable to have a small supplementary journal as 
shown in the figure, in order to meet unexpected lateral pres- 
sures. In such half journals, provided the unused side of the 
material is proportionally increased, d is independent of P, 
and p only is to be considered. We have for 







Wrought Iron. 


Cast :rcwi. 


Steel. 


Po 


= 


S500 


4250 


14,220 


p 


= 


6700 


3340 


11,160 



Example : For a pressure P"^ 220,000 lbs., acting in a constant direction 
upon a slow moving journal of wrought iron, we have from (93) d = o.oi-j 
s/'z-zo^ooo = 7.97", say 8", and / = 4"; for a fork journal, according to (98} 
f/ = 0.0121 \/22o,ooo ^ 5.67", and / is the same; for a multiple bearing with 
eight parts on a side d = 0.35 X 5.67 = i.gS", say 2", and a total length 7 = 2 
X 16 -= 32". If now we take for a half journal the same conditions and make 
d == 2", we get /^ 2 X 8 ^ 16". We may, however, make d .= 1.5", iu which 
case / = T-*s " X 16 ^ 21 28". The journal friction will in this case be 5 that 
of the overhung journal, f^s that of the fork journal, | that of the multiple 
bearing journal, which latter is nearly 60 per cent, longer. 

Au application of this form of journal will be seen in Fossey's 
Coupling. Woolf has also used it ou the cast iron crosshead of 
a large pumping engine.^ 

The principle of the half journal may be seen carried to its 
extreme limit in the knife edge bearings of weighing machine 
in which the friction is reduced to a minimum. The superficial 
pressure upon these very small surfaces is correspondingly 
high, ranging from 15,000 to 150,000 lbs. per square inch. The 
hardened steel edges and bearings seem to be able to stand these 
pressures without injury. f 

? 96. 

Friction of Journai,s. 

New journals show greater frictional resistance than those 
which have worn to a good bearing. 

At first the journal only comes in contact with the metal of 
the bearing in a limited number of spots until after a moderate 
amount of wear the superficial pressure is distributed over the 
projected area of the bearing, giving the value ofp, as indicated 
in § go.J 

For a diameter d, and load P, for a cylindrical journal, whose 
ccefficient of friction =y, we have for the initial force P, which 
the resistance of friction holds in equilibrium, 
for new, unworn journals 



F-- 



-fP, 



and for smoothly worn journals 



F-- 



~fP 



The reduction in frictional resistance is equal to — ; or about 

o.Si times less in a smoothly worn bearing than in a new one. 
The actual value of F is, however, greatly dependent on f. 
This, however, is not only dependent on the lubrication and 
condition of surfaces, as according to the theories of Morin and 
Coulomb, but also upon the superficial pressure p, and speed of 
rubbing surfaces v. \ 

Additional researches upon this subject are yet greatly to be 
desired. 11 



Rennie's experiments with cast iron journals in bronze bear 
ings, with copious Uibrications : 

When /> = 3.2 175 315 492 668 739 
/" = 0.157 0.225 0.215 0.222 0.234 0234 

no account being taken of v, in these experiments. 

Hirn experimented with cast iron ou bronze with full lubri- 
cation, the value of v being equal to 335 feet per minute : 

When / ^ 3 5.26 7.54 9.71 12 

y= 0.0376 0.02II 0.0226 0.0199 0.0183 

and these experiments showed that for small values of/, _/ 
diminishes as, p increases. 

Hirn also found that \i p remained constant, and equal to I2 
lbs., that when 

Z' = 92 164 184 275 327 335 367 
/^ 0.0086 0.0121 0.0128 0.0165 o.otSi 0.0183 0.0191 

thus being at all times quite small, but still constantly increas- 
ing with the increase of velocit}'. 

Morin's researches gave with pressures of 14 to 20 pounds 
per square inch, values of y, from 0.05 to o.ii for journals lubri- 
cated with oil, and from 0.08 to 0.16 when lubricated with 
grease. 

The following results were obtained at the Royal Technical 
Academy from experiments after Morin, upon Clair's apparatus. 
The journal was of wrought iron in brass bearings, freely lubri- 
catecl with oil . 

First Test. Second Test. 

Bearing Surface ....... 12.800 sq. mm. 128 sq. mm 

Total pressure /■ 16.5 kilo 16.5 kilo 

Pressure pr. sq. mm 0.00129 kilo 0.129 " 

Observed friction 1.25 kilo 2.65 " 

Coef&cienty 0.076 0.160 

The author's experiments with an apparatus resembling a 
Prowny brake with surfaces of wrought iron on bronze with 
good lubrication and velocities of 30 to 35 feet per minute, gave 
the following results : 

P= 50 122 192 335 484 624 711 
y^o.ogoo.oS7 0.095 0.118 0.171 0.184 0.180 

Here the value of y was doubled, while/ increased 15 times. 
If/ remained constant and equal to 470 lbs. we have 

for z' = 79 14.17 34.64 55.1 

y^ 0.222 0.210 o 191 0.167 

In this case the coefficient of friction diminishes for an increase 
in the value of n, contrary to the results in Hirn's observations, 
the value of/ being above 40 times greater than Hirn used. 

These latter results appear to be more in accordance with 
Morin's, in that the friction of rest is greater than the friction 
of motion, and hence for small velocities the friction should be 
greater than with higher velocities. This law appears to hold 
good only between certain limits for v, either side of which J 
increases for increasing velocity. Hirn's researches lay beyond 
these limits. Those of the author are, only preliminary to a 
fuller series of observations. 

The following table give some results of the wear on boxes 
of various kinds in railway service : 



* See Tredgold, " Cornish Bumping I^ngines." 

I In large track scales, pressures as high as 425,000 lbs. per square inch 
are found upon bearings less than j'l" wide. The knife edges on the large 
Werder Testing Machine at the Royal Technical Academy are 360 mm. long, 
and sustain a maximum pressure of 100,000 kilograms, or 277.S kg. per mm , 
or at i mm., in width is equal to 556.6 kilograms per square millimetre, or 
Sio.ooo lbs. per square inch, and this pressure has been sustained without 
apparent injury. 

X See Reye, Theorie der Zapfenreibung, Civ. Ing- VI., i860, p. 235 , also 
Grove, Trag-uudStuzzapfen, Mitth. d. Gen. Vereins fur Hannover, 1876. 

§ See Hirn, Etudes surles frottements medints. Bulletin von MUlhausen, 
1854. p. 1S8, also the researches of Reunie, Sella, Bochet, and others. 

\ Engineer, Nov., 1873, p. 312, contains a brief, but valuable discussion 
upon the action of railway axles in their actual conditions of operation. 
The following abstract gives the results ; 

The brasses were all , poured from the same crucible and consisted of a 







Distance. Km. for 


Wear on 4 boxes in 




Kind of Alloy. f 


a wear of i kilo- 


grammes for 1000 






gramme from 4 boxes. 


Kilometres. 






Kilometres. 


Grammes. 


I. 


Gun Metal S3 Cu. 17 Sn. . . 


90,390 


11.06 


2. 


" " S2 Cu. 18 Sn. . . 


99,900 


10.01 


^• 


White Metal 3 Cu. 90 Sn. 7 Sb. 


72,280 


13.83 


4- 


" sCu.SsSn. 10 Sb 


8S,i45 


11-34 




Lead Composit'n 84 Pb. 16 Sb 


81,280 


12.30 


t 


Phosphorbronze 


439,200 


2.33 


7- 


Parsons' White Brass . . . 


385,275 


2.60 


8 


Dewrance's Babbit Metal . 


637,679 


1.57 



mixture of 7 parts copper and I part tin. They all worked under the same 
car and all had the same lubrication. In running 28000 miles the losses were 
as follows : 



1. d = 



Journals. 




Boxes. 


3j. /"Si, 
3J, "-6*, 
33, " = 7", 


loss = 3'," ; 


loss = 5 lbs 
" — 3 '• 

" _2j" 



Taking the journal load as 1 
612, 554 and 427 lbs. 



[,000 lbs., the value of/ in 'the three cases is 



\ Nos. t to 6 are from the work of Dr. Kunzel on Bronze bearings, Dresden, 
1875. The others are from The Engineer, Vol. 41, 1S76, pp. 4 and 31, all be- 
ing given in metric quantities as readily comparable. 



THE CONSTRUCTOR. 



65 



B. THRUST BEARINGS. 

I 97- 
Proportions of Pivots. 

A thrust bearing which is formed on the end of a shaft and 
bears the pressure upon its sectional area, is termed a pivot. 
For ordinary cases these are made in the form shown in Fig. 
178. The pressure p is uniformly distributed over the area of 
the end of the shaft, and the velocity is proportional to the dis- 
tance p of any given element from the centre. A small oil 
chamber of a radius i\ is formed in the middle of the bearing. 

If the outer radius is ;-„, we have 



/' 



0-5^ (n+n ) 



and for the elements on the outside radius 
0-5/ (''i+''o) 

In the formulse for a uniformly distributed pressure /, we 
have taken t\ = J Va and the two diametral oil channels are 
made of a width = ^Jj d. We then have for a given load P: 

P=S,i6pd'> (loi) 

In order that there may not be too much wear for fast run, 

ning bearings (see § 90) we may take/ = ~, and have for high 
speed pivots : 



/'=Si6rf^ — 



(102) 



Alternating pressures do not occur in these bearings and 
need not be considered. The value of a may be taken for 
wrought iron on bronze as = 75. 

Bearings of lignum vitfe running in water may bear loads of 
1500 pounds per square inch even at high speeds.* 

The following formula and tables will serve for the propor- 
tions for end pivots : 

FoRMuiv.5 FOR Pivots (103) 

Wro't Iron or Steel Cast Iron Iron or Steel 
on on 

Bronze. Lignum Vitae. 

700 

0-05 \/p 

350 ^ 1422 _ 
O.C7 ^p 0.035 ^p 

/=1422 

(!'= 0.035 v/7> 



on 
Bronze. 



{ p^ 1422 
Slow movmg Pivots | ^^^ 035 ^/-^ 

f/ = 70o 
M=:or<iSo ^ ^ „ „- /— 



n> 150 



= 75 _ 

= 0.004 V'p;, 





Ft<at Pivots. 




d= 


0.035 \/~p 


0.05 \/p 


0.07 ^.p 


I 


816 


398 


204 


1-25 


1275 


622 


319 


1.50 


1836 


S95 


459 


1.75 


2500 


1219 


625 


2.00 


3265 


1592 


816 


2.25 


4132 


20l6 


1033 


2.50 


5102 


248S 


1275 


2.75 


6173 


30II 


1543 


3.00 


7347 


349+ 


1836 


3-25 


8622 


4205 


2155 


3-5° 


1 0000 


4S77 


2500 


3-75 


"479 


5599 


2869 


4.00 


13061 


6370 


3265 


4.25 


14745 


7192 


3686 


4-50 


16530 


8063 


4132 


4-75 


18418 


8983 


4604 


5.00 


2049S 


9954 


5102 


5.25 


22140 


10974 


5535 


5-50 


24694 


12044 


6673 


5-75 


26990 


1 3 164 


6747 


6.00 


2938S 


14334 


7344 


6.25 


31890 


15630 


7972 


6.50 


34490 


16900 


8623 


6.75 


37190 


18220 


9298 


7.00 


41690 


19600 


lOOOO 



Example I. A craue in the harbor of Cherbourg carries a load of 33,000 lbs. 
ou an eud pivot 6f diameter. Adding its own weight of 6600 lbs. gives a 
value P= 39,600 lbs. This is a slow moving pivot and we have from the 
table for this load a diameter between 6:f and 7.00". A similar crane of 
4000 pounds weight and 20,000 pounds load has d^6-^j.", while the table 
would give about 5A". 

Examples. A driving shaft making 100 revolutions per minute, with a 
load of 2200 lbs., should have, by the table, a diameter of about 2|". 

Example 3. A turbine, making 200 revolutions per minute and 3080 lbs 

load, sliould have a step, according to (103), of 0.004 \/Pn = 0.004 \/3o8q X 200 
= 3s". 

The length of journals in the case of such pivots is usually 
made from i to 1.5 d, its value being sufficiently great to pro- 
vide for the lateral pressure. 





k....p.-.->I 



Fig. 278. 



Fig. 2S0. 



There is a general tendency in machine practice to use 
smaller diameters for pivot bearings, f in order to reduce the 
resistance of friction. 

In order to reduce the effect of higher speeds upon pivots 
bearing heavy pressure a series of disks is often used. If, in 
Fig. 279, the number of plates between the eud of the spindle 
and the step is i, 2, 3, 4, . . . i, we have for the proportion of 

turns between each pair of surfaces yi, ji, ji, times n. 

i-f/ 

This device has been used for steps of turbines, mill spindles, 
etc., by Escher, Wyss & Co., Reiter and others. But few ex- 
amples now remain of this firm for the thrust bearings of 
screw propeller shafts ; the disks bound together and were 





Fig. 279. 

overheated and injured. So far as experience indicates, such 
thrust bearings are capable of standing pressures of 1400 pounds 
per square inch or even more. The important point to be con- 
sidered is, therefore, the reduction of the superficial pressure /. 
The use of other materials than iron, wood or bronze, and 
their substitutes, such as white metal. Babbitt metal, etc. , has 
often been attempted. The subject of wooden bearings will be 
considered hereafter. Besides the use of hardened steel, which 
is of small value for great pressures, such bearings have also 
been made of stone, glass, J or hard burned clay,J but none of 



* Penn has used lignum vitae bearings with pressures oi p = 
pounds. (See Burgh.) 



■ 7000 to Sooo 



t At the establishment of Gruson, in Madgeburg, a boring mill is made 
with cast iron spindle in cast iron bearings, with a superficial pressure of 
more than 20,000 pounds, without ill results. 

X Bearings of glass have been used for more than twelve years atthe works 
of E. Acker & Co., at Graggenan, near Rastatt. These bearings are very 
durable and cheap and require but little lubrication. 

g Shown at the Exposition of 1867, by Leoni, of I,ondon, with good results. 



66 



THE CONSTRUCTOR. 



these materials have come into general use. Girard used a 
pump to keep a film of water between the friction surfaces, and 
atter deductmg the power to operate the pump showed a very 
light resistance.* A similar device was shown by Girard at the 
Exposition of 1867, in which the water jet was 'operated by a 
blast of air. This apparatus was rather of the nature of a 
scientific apparatus, than as a practical application. There 
were also exhibited journals which ran in bearings in which 
water was inclosed. t The experience of general practice, how- 
ever, shows that the ordinary forms are sufficient, without re- 
quiring the use of any of these complicated devices. 

? gs- 
Friction of Fi,at Pivot Bearings. 
If a flat pivot bearing with annular bearing surface, as in Fig 
278, has an inner radius ')\, and an outer radius ?-„ with a load 
-r, we have for the tangential frictional resistance 



F= —fP 
3 



I — -' 



(103) 



in which f is the coefficient of friction. For rapidly running 
pivots we have 



F= -Lp 



' + 7-V 

'0/ 



(104) 



The second value is rather less than the first, since, from the 
previous proportions r-^ = \ r^, which gives for running pivots 
p= '^fP, and the ratio of the two values is as 7 to 6, while if 
i\ = O, it is as 4 to 3. For values of _/"see § 96. 

ro 3) 



Example. In thecraneof example i, §97, P=39, 600 lbs. ro = 3H"- 
/■^o.is. This gives in {104) 



j^= 0.075- X 39,600. 
3 



= 39§olbs. 



The force required to overcome this resistance, if acting: at a lever arm 40 
inches from the axis would be — ■ 



3960 X 3-° 5 
40 



322 lbs. 



i 99- 
Coi,r,AR Thrust Bearings. 

The use of collars to receive thrusts on hocizontal bearings is 
similar to such use on vertical shafts, and a form is shown in 
Fig. 280. In this case the inner diameter 2r, cannot be less 
than the diameter D of the shaft. It is best to make it suffi- 
ciently greater to permit a small oil channel to be used as 
shown in the figure, and oil ways should also be cut in the 
bearing surface. 






Fig. 281. 



Fig. 282. 



Fig. 283. 



If ^0 — ?', is made ftie same as before, good proportions will 
be obtained, although the rubbing surfaces will move at a some- 



what higher velocity. For this reason such bearings are not to 
be recommended when high values of P must be carried. The 
resistance of friction may be calculated by the formuteof the 
preceding section. 



Mui,Tipi<E Coi,EAR Thrust Bearings 

Thrust bearings are frequently made with a row of collars on 
the shaft as shown in Figs. 281-283. If the collars are similar 
the pressure may be taken as distributed uniformly among 

them. If / be a constant value we have for m collars but — 

part of the value is given by (ro4) for each collar, although the 
total frictional resistance will be the same, being the sum of 
the resistances of all the collars. Nevertheless the results of 
experience, especially with screw propeller shafts, shows the 
necessity of making ni large in order to keep the pressure * as 
small as possible. This is due to the fact that heavily loaded 
shafts give, according to (104), so great frictional resistance as 
to case excessive heating and consequent injury, and experi- 
mental researches have shown the value in reducino- p and 
consequently/ The best values of/, lie between 40 to 80 
pounds per square inch. When bearing of this kind is placed 
at the end of a shaft it may be reduced in diameter as shown in 
Fig. 283, and in such cases p may be made somewhat greater 
even as high as 350 pounds, but in such a case there is a great 
tendency to heat. 

Example I. Screw Propeller Engine, by Indret. Thrust on shaft 39,600 
lbs., w— sslbs. 2ri>=Z)=i5". Breadth of collars =* = m—r, =2" Num- 
ber of collars ^=9. /^o n 2 . iMum 



Here/. 



39600 



= 40 lbs. 



The velocity z/, at a radius 



9fXi7X2 " 
-275 ft. This gi%'es in (104) takingy"= 



F^ 39600 



and-the friction horse power 
HP = 



9-5 



I = 3465 lbs. 



3465 X 273 

= = 29 H. 

33000 



Example 2. 
^8. 2 ?-i = i? . 



Turbines on the Rhine at Schaffhausen. 
= 9". Collar width <5, =. ro — rj = i s.^", m = 

30800 



/'^ 30800 Its. « = 



This gives/ ■= 



V — 133 feet. 



9 IT X 10,625 X l.6zs 



= 63 lbs. 



_ C.I / 

F= 30800 ( 



9 
12.2^ 



2664. 



H 



2664 X 133 „ 

P. = = 10.7 H. 

33000 



Example 3. Girard Turbine at Geneva. J 
P= 33,000 lbs. w = 16, 2 ri = ZJ ^ 9.8'' 
b = Tq — rii ^ %", nt =12 

3^. coo 
This gives/ ^'^' 



12 TT X 11.17s X 1.375 ~'' 
1/ = 46.7 ft. From (104) we get F— 2970 and the friction horse-power is 

!97o2<j^.7^ 
33.000 



H P __ ^' ^ ^ ' 



iH.F. 



Example 4. I^angdon lays down the rule that for collar thrust bearings 
of screw propeller engines there should be % square inch of surface for 

every indicated horse-power. j' ^^ «■ ^^-1 

th.e ship 



1-. V. .^..^u.u ^^ yiov^iiai.^ luLii ui biiriace lor 
If A'= the horse power and c the velocity of 



This gives p ^ 



N=-^^ 

33,000 

33,000 X P 44,000 
0.7s Pc c 



add it c ^ 1000 ^. ser minute, p -= 44 lbs. 

Exarapl-e5. A large centrifugal machine by Langen & Sons, in Cologne, 
has a collar step of the following proportions ; 



P= 4400 lbs. n = 800, 2 ri = i", 2 ^o 



= 1.57" w = II 



p=- 



11^(0.782-0.52) 



-' 366 lbs. 



which is an excessive pressure, liable to cause heating, and demanding 
most careful lubrication. In this case v = 275 and taking/"= o.i we get as 
before 

1,60 X 271; 
=- — - — ~ 3 //. P. 
33,oco 



i^= 260 and-^. P. 



*See Armengand, " Vignole des Mecaniciens," p. 139. 

t Exhibited by Jouffray. See Armengand " Progr6s de I'industrie & I'exp. 
universelle," Vol, I, PI. 8. 



jOppermann, Portefeuille econ des machines, Vol. 17. Also Engineering, 
1872, Vol. 14, p. 233. 

2 See Burgh. 



THE CONSTRUCTOR. 



67 



In all these examples the co-efficient of frictiony" has been 
taken = 0.1, and for the moderate pressure of the first three ex- 
amples a lower value might have been taken. The examples 
will suiEce to show the importance of the selection of a suitable 
value for/, and other cases will be examined in § 122. 

I lOI. 

The Compound Link as a Thrust Bearing. 

In the previously examined cases it has been the object of 
the various plans to reduce the journal friction to a minimum, 
but there are sometimes occasions in which it is desired to give 
a journal a definite amount of frictional resistance, without 
danger of its sticking fast, so that it may be rotated with a 
moderate force, and may also be readily clamped in any desired 
position. This may be accomplished, for example, by a thrust 
journal made in the form of a truncated cone. If the radii of 
the large and small ends are respectively Tf, and r^ and the half 
angle a, we have for the force F, instead of (104), 



/ 
2 



P 






(105) 



and by varying the angle a, may give any desired value to F.* 
Very acute pivots sometimes bind in an injurious manner, 
and hence the increase of /^cannot be carried to an extreme in 
this way. Clamping of this sort may better be accomplished 



Cz 
Cz 

c- 

c= 



ID 



zD 



i i ! i i 

!<— -2ro— »i 

Fig. 2S4. 

by the use 'of compound bearing surfaces, so arranged as to 
press on each other, as shown in Fig. 2S4. Each plate then 
transmits the axial pressure to the next. Ifm is the number of 
contact surfaces, the friction at the radius ?'„ of the bearing is 
found by an analogous equation to (104), 



F= m^P 



0+^) 



(106) 



Example. Let F := P, and lety"== o.i 
20 



■ whence, if i\=^ yi ?'„, m= 13 



1 + -^ 
''0 

This arrangement has been used by the writer with success in 
many parts of machines where a clamp was desired. Formerly 
the j oints of dividers were made with four plates at the pivot. 

? 102. 

Attachment of Journai^. 

" When a journal cannot be made in one piece with the rest of 
the shaft, various methods of attachment may be used ; such 
devices are mainly necessary in fitting iron journals to wooden 
shafts, as for water-wheels. 



* Applications of this principle may be seen in' the spindles of astro- 
nomical and surveying instruments. Formula (105) may be used to de- 
termine the friction of stop-cocks. 



In Fig. 2S5 is shown a form of attachment in which a cross 
anchor piece is forged on the shank of the journal, and a slot 
mortised in the end of the shaft to receive it. After the journal 




Ma 



Fig. 285. 



Fig. 286. 



is in place it is clamped by driving on the previously heated 
metal bands (see § 62). The angle of taper is jJj-. Fig. 286 is a 
very good form in which the shank of the journal is keyed in 




Fig. 2S8. 



place. In Fig. 2S7 is shown a cast iron journal with two wings, 
arranged to be driven in, and Fig. 28S shows the proportions of 
the same when four wings are used. If three wings are desired 
their thickness may be made equal to j% d. 




Fig. 289. 

Fig. 289 shows a form in which the four' wings are surrounded 
by a conical shell, which is held in place by bolts and anchor 
plates. The shell is sometimes made with keyways cast in it to 
act as a centre for the hub of a gear wheel. 




Fig. 290 shows a very practical form. The journal is cast on a 
plate strengthened by heavy cross arms, and a wrought iron 
ring is shrunk on, while the whole is fastened to the shaft by 
the four bolts, whose nuts are let into the wood, as shown . 



68 



THE CONSTRUCTOR. 



CHAPTER VI. 
BEARINGS. 

I I03- 
Design and Proportion. 

The mechanical devices by which the journals of shafts and 
axles are carried are called bearings. A complete bearing may 
be divided into three portions: i, the boxes; 2, the body or 
frame ; 3, the connecting parts. 

The various forms may be divided according to their uses 
into the two main classes : 

A. Bearings for Lateral journals or Lateral Bearings. 

B. Bearings for end-long pressure or Thrust Bearings. 

Under these classes the principal distinction is to be made as 
to the side on which the bearing is to be supported. If we 



--,T 





construction of bearings, and the following example will show 
its use : * 

The poles O, O^, O^, Fig. 294, are used for the journal diam- 
eter d ; the poles P, P^ and P^, for those dimensions which de- 
pend on the modulus rfi= 1.15 rf -|- 0.4". This gives di=0. 



Fig. 291. 



suppose the journal to be inclosed in a cube 1.2 . . .8, Figs. 
291, 292, we have for lateral bearings 

A Pillow Block, when the base lies in i, 2, 3, 
A Wall Bearing, " " " " i-S or 2-8, 
A Front Bearing, " " " " i-6or4-7, 
A Hanger, " " " " 5-7. 

For Thrust Bearings we may have Foot Step Bearings, Wall 
Step Bearings, or Hanging Step Bearings. 

Especial care is to be taken for the equalization of wear and 
for efiicient lubrication, and these points affect mainly the 
boxes. 

The examples which follow have only been selected from the 
vast number of forms to show typical cases. 

The dimensions are based upon a proportional scale. As the 
unit for the thickness of the brasses we have ^=0.07 d -\- yi", 
d being the bore of the boxes, and volues of e are given in the 
second column of the table in \ 91. The modulus for the body 
of the bearings is : 

i/i = i.i5(/+ 0.4" .(107) 

ZA.— LATERAL BEARINGS 

I 104. 

PlUOW Bl,OCK 

In Fig. 293 is shown a form of pillow biock suitable for jour- 
nals from ij4'' to 8". The proportions of the body and cap 
are based on the modulus d-^ (see 107), with the exception of 
the oil cup on the cap, which would then be rather too large for 
small bearings, in which it is made in length equal to the 
width of the cap, and in width equal to 0.7 d-^. 

The length of the boxes is dependent upon the length of the 
journal, which, as discussed in \ 90, may be 1.5 d, 2 d, etc. 
For the form shown a good proportion is 1^2 d, the projecting 
portion of the boxes being governed by the proportion of 
length to diameter adopted. 

The bolts for the base plate are made somewhat heavier than 
those for the cap, as they are screwed up much tighter, and 
they are often made with special heads to fit a separate sole 
plate as shown m Fig. 294. The ends of the base are given a 
bevel in order to permit the use of side keys. The coring out 
of the sole plate reduces its weigh and also simplifies the ma- 
chine work. The spaces between the cap and the body of the 
bearing are filled with slips of wood so that the cap bolts may 
be tightened without binding the shaft. In cases where the 
load is great, the pressure alternating, the joint is closely fitted 
without spaces, and if wear in the journal is to be taken up the 
surfaces are filed down. 

I 105. 

Proportionai, ScaIvE FOR Pillow Blocks. 

The proportional scale may be used to great advantage in the 




Fig. 293. 

when d = -:-^ — n:_L = 0.34", hence P must be placed when 

1.15 
the vertical space between the rays Oa and Ob is equal to 
— 0.34''. The intersection of the rays from O and from P, by the 
ordinates I, II, etc., give the dimensions of the corresponding 
sizes. The dimensions of the boxes must be obtained from 
another pole, as they depend upon another modulus. This 

modulus is e^.o-j d-\- }i'' and becomes = O, when d 

= — 1.78". ■ The poles E and E^, therefore, are placed on the 



o.i^ 
.07 




Fig. 294. 



vertical line on which the distance <;' i' equals 1.78'' 
the oil cup in the cap the width is : 

0.25 di -f o.a" = Q.\" + 0.25 (0,4" + 1. 15 d) = 
= 0.29 d -(- 0.5" 
= 4.16 (,07 d-Y yi") =4.16^ 

Hence E is also the pole for the oil cup. 



For 



* The firm of Escher, Wyss & Co , in Zurich, have used the proportional 
scale very well for designing bearings, both in determining the geometrical 
proportions throughout and also by the excellent method of a single pole. 



THE CONSTRUCTOR. 



69 



I 106. 

A^ARious Forms of Journai^ Boxes. 

It is often found convenient to give the boxes of a pillow- 
fclock other forms than those of the preceding illustrations, as 
for example octagonal, as in Fig. 295, or cylindrical, as in Figs. 
296 and 297. The last two forms are suitable for bearings in 
lathe headstocks, and in such cases the boxes are kept from 



...!.d,_.......J 



the form shown in Fig. 294, \ 105, so that the base may be re- 
moved from the base plate when necessary without disturbing 




Fig. 295 



Fig. 296. 



slipping out of place by the flanges whose width is 2e, as shown 
in Fig. 296, or by projecting pins, Fig. 297, fitting into recesses 
in the base and cap. Each of these forms has its advantages 
and objections, and it is hardly possible to decide which form is 
the most desirable, special conditions being generally present. 
The modifications in the base and cap to admit the forms shown 
in Figs. 296 and 297 are readily made without requiring detailed 
instructions. 

Boxes in which white metal or similar compositions are used 
require special construction, since these materials are not strong 
enough to resist the stresses with the same securit)' as solid 
bronze boxes ; for such bearings a cast-iron or bronze shell is 
made, in which a lining of the softer metal can be poured, as in 
Fig. 29S'. In such cases the shell should be cleaned with acid 
and tinned before pouring the lining metal. 

Boxes of lignum vita; (see §| 97-117) must be made of 
simple shape. A convenient shape is shown in Fig. 299, 
which the general form of the bearing may be made. 

In America examples are often found of bearings in which 

Fig. 298. 




Fig. 299. 



Fig. 300. 



the soft metal is run directly into recesses in the base and 
cap. Fig. 300 shows such a bearing as made for the journals of 
fan-blowers and shafting, by Sturtevant, of Boston. The base is 
hollowed out to serve as an oil chamber, and the oil is fed to 
the journal by a wick. The details are shown in Fig. 301. These 
journals are made very long (/ ^= 4^), and hence the superficial 
pressure is small. 

I 107. 

Narrow Base Bearings. Large Pii,low Bi,ocks. 

It is often desirable, when space is limited, to make bearings 
with narrow bases, and this may be done by making the cap- 
bolts with collars as shown in Fig. 321, and also Fig. 312. This 
permits the holding down bolts to be dispensed with, and space 
saved. Such collared bolts are also used for pillow blocks, 
which are subjected to both upward and downward stresses, 
since the boxes are firmly bound together (see \ S8). Fig. 302 
shows a form of pillow block for journals of S to 12 inches in 
diameter. It is made with four cap bolts and four base bolts, 
by which it is secured to the base plate. The base bolts are of 




XIX 



Fig. 301. 

the soHJity of the latter. The body of the pillow block is cored 
out to a greater extent than in the previous form, and when 




Fig. 302. 

the journal is used for a crank shaft, or is subjected to jarring 
strains, the cap bolts should be provided with jam nuts, or some 
of the other forms of securit}-, such as is shown in \ 8$. 

{. loS. 

Pinow BivOCK WITH Adjustable Bearing. 

In man3' cases it is only necessarj' to adjust the height of 
pillow blocks from time to time by inserting liners beneath the 




Fig. 303. 

base, but in some situations it is desirable to provide a special 
means of obtaining such an adjustment. In- Fig. 303 is shown 
such an adjustable bearing for use in screw propeller shafts. 

The body of the bearing is not bolted down, but rests solely 
by its weight upon the wedge system, by means of which it can 
be raised or lowered as may be found necessar}'. The upper 
box is provided with flanges through which the cap bolts (omit- 
ted in the illustration) pass. The lower box is lined with white 
metal, which is poured into the recessed bearing. 



"jO 



THE CONSTRUCTOR. 



i 109. 
Adjustable Pillow Blocks. 

Many attempts have been made to arrange the boxes in a 
pillow block so that they may be self-adjusting and so adapt 
themselves to various positions, which the journal may assume 
and secure for it at all times a full bearing and support.* For 
this purpose, among other methods, the plan has been adopted 
of making the boxes with central spherical portion fitting into 
corresponding recesses in the body of the pillow block. This 
form of bearing has been widely introduced in America by 
Messrs. Wm. Sellers & Co., and adapted to a great variety of 
positions. 

Sellers has always urged the desirability of the principle of 
keeping the pressure between journal and bearing at a mini- 
mum, f This practice permits the use of cast iron boxes, for 
which a pressure of not more than 15 pounds per square inch is 
used. J 

The use of moderate superficial pressures is most practicable 
in the case of bearings for line shafting in which the propor- 
tions may be made such as to give but light pressure. This ad- 
vantage will be seen on reference to g 92. | 

Fig. 304 shows Sellers' form of pillow block. The cast iron 
boxes are made with a spherical enlargement in the middle, 
which is held between corresponding recesses in the cap and 
base. The boxes are prevented from revolving by the hollows 




Fig. 304. 

in the sides which receive the bodies of the cap bolts. Three 
openings are made for oil or grease and two drip cups, which 
are cast on the base plate, serve to receive the superfluous oil il 

The modulus upon which the proportions of this bearino- are 
based, is not that given in (107), but the following -^J "^ 



n 



= 1 .4 (^ -f o. 



(loS) 



The length of the boxes =40'. The shape adopted by Sel- 
lers shows the care in modelling which is characteristic of the 
American designs of engineers. The Sellers' bearings have 
been used to a considerable extent in Germany. 



* Various designs have been made by Bodmer at Manchester Schonherr 
at Chemnitz, Stehelin at Thann, and Zimmerman at Karlsruhe.' 

t Treatise on Machine tools, etc., as made by W. Sellers & Co. Philadel- 
phia, Lippincott. 1S73, p. 161. ' 

jAs an example of the performance of these cast iron bearing's Sellers 
cites a bearing which had been in service forsixteen years and iu which the 
lower box was not yet worn to a polish over its entire surface. The shaft 
made 50 revolutions per minute and was 4}^ in. diameter, and carried near 
the bearing a 72 in. pulley, of 20 in. face, transmitting 52 horse-power In 
other examples it is shown that after a year's use the tool marks were' still 
visible. The small superficial pressure does not force the oil out and 
hence the journal is carried on a film of lubricant The consumption of 
oil is very small, and Messrs. Sellers State that a shaft inakin<^ I'o revolu- 
tions per minute consumed but 2'/, ounces of oil in six months. 

g As an example of the impracticable results which would follow from au 
attempt to obtain such light pressures to overhung journals we may take 
Sellers' value of 15 pounds per .square inch and apply it to an example If 
>"= 17,600 lbs., we have for a wrought iron shaft with a constant direction to 
the pressure, from the table in J 91, d^ 4'', 1—6". If p _ 15 lbs. we have 
from formula {90) _ =11.9 and from (89) we have rf — n.="; hence 2 = 
II. 2 X ll-g = 133" ! ! 

II Sellers recommends a mixture of tallow and oil, which becomes more 
liquid should the bearing grow warm. 

if See Berliner Verhaudlungen, iS76,"p. Sg. 



Another form of adjustable pillow block is shown in Fig. 
305. This is used by Sturtevant in some of his fan blowers. In 
this case the ratio of / to rfis very great (see example 4, I 91). 
The adjustability is obtained by pivoting the bearing A upon a 



A J 




Fig. 305. 

cross bolt B, which passes through the cheeks of the pedestal 
also ; the latter being adjustable about the axis BC. The bear- 
ing is lined with white metal, and the end thrust is taken up by 
a block of lignum vitse. If an adjustment in the direction A A 
IS required, the bolt C may be loosened and the required move- 
ment made. The provision for lubrication is especially note- 
worthy both in the manner of supply and in the collection of 
the overflow. 

i no. 

Bearings with Three-Part Boxes. 

In horizontal steam engines and in similar service, the 
pressure upon the journal is thrown first on one side and' then 
on the other, while at the same time there is a constant vertical 
pressure, such for instance as is due to the weight of a fly 
wheel. Attempts to remedy the tendencv to overwear by mak- 
ing the boxes inclined, have proved but a partial remedy, and the 
best method of construction in such cases is to make the box 
in three parts, one of which receives the constant vertical 
pressure, while the other two provide for the backward and for- 
ward thrust. Such a bearing is shown in Fig. 306. The modu- 
lus (/i = 1.15^-1- 0.4". The bottom box rests on two wedges 
which are tapped with screw threads and can be adjusted and 
locked at any desired point by the bolts shown. The side boxes 
are each held up by two steel set screws ; a wrought iron plate 
being interposed between the screws and the boxes. If it be- 



THE CONSTRUCTOR. 



7f 



comes necessary to remove the side boxes the cap is first taken 
off, and the iron plates taken out, when the boxes can be sepa- 
rated far enough from the shaft to permit their removal without 



those cases in which an alternating up and down pressure is 
combined with a constant lateral pressure. The latter would 
not be provided for in an ordinary pillow block, but here it 




Iktk! ftMliJij, ilib.il, t^iaitHnA n p m % i itjutiltiltilii "!:^' 




Fig. 306. 

interference with the shaft. The body of the bearing is in 
creased in width in order to provide for the ^increased lateral 
pressure. 




Fig. 307. 



Another three-part bearing * is shown in Fig. 307. In this 
case there is no vertical adjustment to the lower box — and if 
necessary it must be raised hy packing underneath. The side 
boxes are set up by wedges which are adjusted by set screws 
through the cap. Each wedge carries a screw on its upper end, 
and the nuts for these screws are fitted so as to revolve in the 
cap, being turned by a wrench on the hexagonal head, and 
then clamped in position by the thin jam nut shown. The 
heavy inclined ribs stiffen the body of the bearing to resist the 
stock and thrust of the piston. It is often convenient (as in 
the case of the original of the figure) to cast the body of the 
bearing in one piece with the bed plate of the engine. 

A third, and simple form of three-part bearing (by Schultz 
Brothers in Mayence) is shown in Fig. 308. It is suitable for 





Fig. 308. 

is taken up by the small side box. This form is suited for 
small vertical engines in which the pull of the belt is toward 
one side. 

'i III. 

Pedestal Be-4rings. 

Bearings which are not placed directly upon a base plate, but 
are raised upon feet or pedestal are called pedestal bearings 




8=6.6-- 



FlG. 309. 



* From a steam engine by the Soc. Fives- Lille in Paris. 



That shown in Fig. 309 is similar to the one in Fig. 293, placed 
upon a pedestal. Such pedestals vary greatly both in form and 
height. The width of the foot is made equal to the height of 
the journal in the form shown, which gives the base and the 
legs a sufficiently slender appearance. 

I 112. 

W.\LIv BE.ARINGS. 

The wall bearing shown in Fig. 310 is the same as shown in 
Fig. 293, with the addition of the bracket. The base here is 
placed at right angles to the joint in the boxes and parallel to 
the axis of the bearing, the whole being made in the bracket 
form shown. 

The cap and the boxes are of the same form and proportions 
as for a pillow block for the same size journal. The bolts may 
either be tapped into the body of the bearing, or made as stud 
bolts, using the forms shown in Figs. 225 and 226 ^ S3, with 
key. 

For larger sizes the opening in the plate should be surrounded 
with a rib of a thickness cirfj and width = 0.4^/,, the latter 
being measured in the direction of the axis of the journal. 

Fig. 311 shows an adjustable wall bearing by Sellers. In this 
case the cast iron boxes are somewhat lighter than for pillow 
blocks and are made with a cylindrical cross piece in the 
middle, in which the spherical seats are placed. The especial 
feature is the method by which the vertical adjustment is made. 
The two plugs which support the boxes have cast upon them a 
very shallow screw thread, and the nuts in the sockets have 
also their threads cast in them. The thread only extends along 



72 



THE CONSTRUCTOR. 



a portion of the length of the plugs as shown, in order to per- 
mit securing them in position. This is done by the two self 
screws which clamp them firmly in their places. 

The opening through the upper plug gives access for the tube 
of a lubricator. 



..J., 




Fig. 310. 



The projection from the wall a is made constant for bear- 
ings forjournals 2" to 4" in diameter and equals (>". The ele- 
gance of the form is noticeable in the principal elevation and 
also in the horizontal section. 

I 113- 
Yoke Bearings. 

The bearings used on vertical shafts may be considered as a 
variety of wall bearings. In situations where the space is lim- 
ited the forms shown are not always convenient, the first, be- 
cause it is not symmetrically disposed about the parting of the 




Fig. 311. 

boxes, and the second, because of the space it requires. For 
this service a compact, symmetrical bearing, whose base is at 
right angles to the parting of the boxes, is often very desirable. 
Such a construction is shown in Fig. 312, and may be called a 
Yoke Bearing. In this case the cap and body together form a 
rectangular 5'oke, in which the bronze boxes are placed in a 
transverse direction. In the illustration the wear can only be 
taken up in one direction, but if it is desired in both directions 
the cast iron block on the right may be replaced by a wedge as 
shown on the left. 



By removing the cap, the wedge and the block can be easi- 
ly removed and the shaft moved sideways to a sufficient extent 
to permit the removal of the boxes. The cap bolts are provided 
with collars forged upon them and serve also to fasten the bear- 
ing in place. The modulus for the dimensions is the same as 
(107), q'i = 1.150' +0.4".* 




l,5cl 



Fig. 312. 

§ 114. 

Wai,Iv Brackets. 

" In Fig. 313 is shown a form of bearing similar to Fig. 293, 
which may be called a wall bracket bearing. The cap bolts 
are inserted from below, which permits their ready removal and 
replacement. If only two bolts are used in the wall plate, it is 




I/6S 




Fig. 313- 

desirable that it should be held from lateral motion between 
wedges, and should also be firmly secured against vertical mo- 



* For such a Yoke Bearing, see Engineers' and Machinists' Assistant, 
London, 1854, PI. I. 



THE CONSTRUCTOR. 



73 



tions by some of the methods given in the following chapter. 
Where it is not practicable to secure it in this manner, four 
bolts should be used. 




^c-0,r)->K- 1.3 >'k-0,5>[<- 0,87->i 

Fig. 314. 

Another form of wall bracket is shown in Fig. 314. It is sim- 
ilar to the Yoke Bearing, and can often be of service, as for 
example in Fig. 350, § 126, although it is not of as general ap- 
plication as the preceding form. The bolts for the cap are 
made with heads, of the ordinary cap screw form. 

Various other wall and bracket bearings may be made b}- 
combination of a wall plate and pillow block in different po- 
sitions, and these may be grouped in the general class of Arm 
Bearings, each form being governed by the conditions of the 
special case under consideration. 

115. 

H.^NGKRS. 

, According to the definition in \ 103 a pillow block by inver- 
sion becomes a hanger, the pressure of the journal falling upon 
the cap box. If the journal is one of wrought iron proportioned 
to bear the loads given in S 91, the bolts for the cap and base 
plate will not be strong enough if determined from the same 




Fig. 315- 



This is 



unit of proportion as alreadj' given for such bearings, 
also true for the cap, and feet of the base. 

For this service, good dimensions may be obtained by using 
for the boxes the previous modulus (/j = 1.15 rf + 0.4", and also 
E as before, and for all other portions the special modulus. 



/?■ 



1.75 rf + 0.4'' 



(109) 



If a pillow block is to be used as a hanger for a neck journal, 
the cap bolts should be increased to such size as would be given 
by the use of formula (109), in which d is the diameter of the 
neck journal corresponding to an equivalent end journal. 

Example : A load of 17,600 lbs. would give, according to the 
table in J91 for a wrought iron journal a diameter of about 4 
inches. If this load is carried on the cap of the bearing we use 
the modulus, 



This gives for the diameter of the cap bolts 7.4 s 0.2 = 1.48"^ 
say I }i,"- A neck journal of 6;4"' diameter to bearthe same 
load would have for its normal unit d ^ 1.15 s 6.75 -j- 0.4 = 
8.15''', which is greater than the preceding value and hence may 
be used safely, even should the full load be carried by the cap. 

Sellers makes a short hanger which resembles in form and 
dimensions the corresponding size pillow block, with the boxes 
turned iSo° and the drip cups cast on the cap instead of the 
base. In most cases, however, a greater distance is required be- 
tween the shaft and the base plate for hangers than is given in 
pillow blocks, for which reason they are best considered as a 
separate form of construction. 

The hanger shown in Fig. 315 is called, from its form, a 
Ribbed Hanger. The boxes are carried in the hook-shaped por- 
tion below, their form bei ng the same as we have already 
shown. The cap is secured with a key and clamped in the de- 
sired position \>y the bolt shown. 

For journals of less than 2 inches diameter, but one bolt need 
be used in each foot, and in such case their diameter is ^ 0.3 d-^, 
the bosses on the plate be altered to correspond. 




Fig. 316. 

In the Post Hanger, Fig. 316, the general arrangement is the 
same as in the preceding form, the principal difference being in 
the frame. The column is made hollow and its internal diam- 
eter = 0.55 d^. For the larger sizes four bolt holes are made in 
the base plate, as shown in Fig. 315. 

Hangers are not generall}- bolted directly to the ceiling 
beams, but to strong pieces, or intermediate timbers, and by 




Fig. 317. 



Fig. 318. 



D'; 



1.75 a' -f 0.4'' = 1-75 x 4" + 0.4'' = 74'' 



varj'ing the thickness of these pieces any desired amount of 
drop maj' be obtained. If the variation is too great to be se- 
cured in this manner a different depth hanger must be used. 
If the building is of so-called fire-proof construction, with 



74 



THE CONSTRUCTOR. 



ceilings of iron beams and brick arches, tlie form of the base 
of the hanger must be correspondingly modified. A practical 
method is shown in Fig. 317, in which hook bolts are used. The 
bolts, which are four in number, pass through sockets cast in 
the base of the hauger, and their method of attachment avoids 
weakening the beam. The base of the hanger is made with 
ledges which fit over the edge of the beam and permit the use 
of wedges on each side. 

The form shown in Fig. 31S, which is due to Fairbairn is in- 
tended to bring the shaft parallel to the beam, while the pre- 
vious form carries the shaft at right angles to the beams. The 
attachment of the hanger both to the beam and the arch makes 
a very secure fastening, but the inaccessibility of the bolt head 
is an objection. In this case also the beam is not %veakened by 
drilling, hook bolts and keys being used, as in the previous 
case. 




»....-.l„6 . « 



Fig. 319. 

I 116. 

• Adjustabi,e H.^ngers. 

EJThe most generally used of the Sellers' adjustable bearings is 
thejhanger shown in Fig. 319. ^ Thel method of holding and 




Fig. 320. 

adjusting the boxes by means of screw plugs is the same as 
shown in the wall bearings, Fig. 311. Especially to be noted is 
the attachment of the drip cup, which may be easily removed 
by withdrawing the small pin with enlarged ends. 



The drop, or distance from base to centre of shaft, ^= a ^ 
3.5 rf/ in the illustration, but in some cases it must be made 
greater. These hangers, like all of Sellers' bearings, show 
very careful modeling and proportioning, which the small size 
of the illustrations can only imperfectly show. 

In Fig. 320 is shown Sellers' countershaft hanger. In this 
form the shaft is put in place from the side, and the amount of 
wear in the boxes is so slight that they are made solid, instead 
of in halves. The cap — which is secured by a bolt, holds the 
box in place, and the drip cup is cast i;i one piece with the 
body of the hanger and provision is made for a drip cock to 
remove the waste oil. 

The illustration shows also the arm for carrying the belt 
shifter. 

Sturtevant uses ball and socket hangers also for the counter- 
shafts of his fan blowers. These are somewhat different from 
the preceding. Fig. 321 shows the boxes in perspective and in 
cross section. The section shows the white metal lining and 
also the arrangement of double oil chambers, which, by means 




Fig. 321. 



Fig. 322. 



of wi eking, keep the journal lubricated." The outer ends of the 
box casting are formed into drip channels, and also receive the 
shoulders on the shaft. These shoulders, as shown in Fig. 
322, run freely in the boxes without contact. The journal as 
shown is on the end of the shaft, and the pressure is so smal! 
that the wear is inappreciable. 

l 117. 

Speci.^l Forms of Bearings. 

In propeller shafts where the screw is arranged to be lifted it 
is necessary to design bearings which are to be entirely im- 
mersed in water. Penn's practice is to line such bearings with 
wood, which has proved especially satisfactory. In Fig. 323, is 
given an illustration of such a bearing as constructed by 
Ravenhill & Hodgson, the diameter of the shaft being about 
19 inches. The body of the bearing is of bronze, the boxes are 
of cylindrical section fitted with strips of lignum vitse set in a 





Fig. 323. 

special lining metal. The pin, projecting from the bottom, 
enters into a corresponding recess in the stern frame, when the 
screw is lowered into place. 

On the Prussian State Railway there have recently been 
adopted two standard forms of bearings for use under cars — one 
form being for bronze, the other for white metal boxes. In 



THE CONSTRUCTOR. 



75 



Fig. 324 is shown details in partial section of the latter form, 
with a few dimensions. The bearing is made in two principal 
parts, the body and the lower portion, both being provided 
with oil chambers having openings and covers to keep out the 
dust. The joint between the two parts is in the horizontal 
plane passing through the axis of the journal, the parts being 



body of the bearing is a filling block, somewhat similar to that 
used in the bearing shown in Fig. 312, arranged so that its re- 
moval facilitates the changing of boxes. 




Fig. 324. 

kept in position by three dowel pins. A wrought iron yoke 
holds the lower portion up to the body of the bearing by means 
of the bolt shown, the head being secured by the internal hex- 
agonal socket shown. 

The white metal lining is cast in the body of the box by be- 
ing poured upon the journal. The inner end of the journal is 
provided with a wooden dust guard packed with a ring of felt. 

As will be seen, lubrication is provided both above and be- 
low. The upper chamber contains wicking and affords a means 
of prompt and copious lubrication in case the journal grows 
hot. The principal source of lubrication, however, is from be- 
low, the oil being wiped upon the journal by a brush, which is 
fed with oil by a wick reaching into the chamber below. The 
oil brush is shown, with its spring holders in the lower right 
hand corner of the illustration. 

In order to permit the boxes to adjust themselves to the 
journal when the axle assumes an inclined position with re- 
gard to the bearing a certain amount of play is given, as is 
shown in the plan view, where the ledges cast upon the 
bearing are made' parallel for a short distance and then diverge 
from below upward from a width of 34 mm. to 42 mm. 

All the dimensions in Fig. 324 are in millimeters, as this is a 
standard Prussian railway journal box. 

This construction is undoubtedly well adapted to meet the 
requirements, but it is a question whether the results might not 
be attained by simpler means.* 

The second form of standard bearing of the Prussian Railways 
differs from the first mainly in the boxes. These are cast of 
bronze with semi-cylindrical projections on the track, which 
enter into corresponding recesses in the bearing, and permit 
the boxes to adjust themselves to the journal. 

The guides for the bearings are given an amount of play 
similar to the previous form, and there is no change in the de- 
tails of the lower portion. 

Fig. 325 shows a form of American axle bearing. This is sim- 
ilar to the older pattern designed by Lightuer t It is only ar- 
ranged for lubrication from below and is designed so as to per- 
mit a box to be removed and replaced in the shortest possible 
time. The body is of very simple form and is cast in one piece 
and a large opening and lid renders it readily accessible from 
without. The box is made of bronze, and between it and the 



* This question of railway journal boxes is an instructive example of the 
importance of constructive simplicity as applied to machine elements. Since 
in the year 1877 in Prussia there were in use 315,000 axles or over 630,000 
boxes. The cost of these represents an investment upon which every 
penny economized in construction foots up an important total. 

t See Heusinger, Schmiervorrichtungen {Lubrication), Wiesbaden, 1864, 
p. 83. 




Fig. 325. 



This filling block, which is sometimes rounded on top to 
provide adjustment, is held between two small projections, but 
can easily be removed when the pressure is removed by use of 
a liftingjack. The change of boxes can be effected in a few 
minutes. 

A brush or pad for distributing the oil is not used, but instead 
the vacant space in the bearing is packed with waste, which 
feeds the oil to the journal. This form of journal box has 
proved very efficient in service.* 

B. THRUST BEARINGS. 

\ iiS. 

Step Bearings. 

In Fig. 326 is shown a form of step bearing for vertical shaft. 
The bearing piece or step proper is made with very obtuse 




Fig. 326. 

point on the under side in order that it may be able to adjust 
itself to the shaft. In order to provide for adjustment in the 
position of the bearing the bolt holes in the baseplate are 
elongated in a cross-wise direction, while those in the bearing 
are elongated length-wise, thus permitting adjustment in any 
direction. 

I 119- 
Wale Step Bearings. 

The following is a modified form of step bearings, and is in- 
tended to be used with the wall plate supported on a key be- 
neath its lower edge ; this key may be made = o.S d-^ deep, so 



* A standard axle and journal box were adopted in the United States in 1873, 
and at that time there were over 1,200,000 axles in service. 



76 



THE CONSTRUCTOR. 



that by its removal the bearing may be taken from under the 
journal, without removing the shaft from its place. 




Fig 327. 

The recess in the step plate serves an oil chamber ; end- 
long wear may be taken up very conveniently by the adjust- 
ment provided by the set screw. 

? 120. 
Independent Step Bearings. 

In many cases, as in examples by Belgian designers, the 
lower bearing of a vertical shaft is divided into two independent 
parts, a pure lateral bearing and a pure thrust bearing. For 
the lateral bearingmay be used a pillow block or yoke bearing 
of one of the forms already described, while the vertical thrust 
is taken by a simple step quite close to the preceding bearing. 
This makes the step bearing readily accessible and also readily 
adjustable in the direction of wear. 

The following example is selected from among a number of 
such bearings. 




Fig. 32S. 

The step itself is made of bronze. This is carried on the 
bluntly coned head of the stout set screw, a steel plate being 
interposed, while the prismatic form of the screw head pre- 
vents rotation of the step. The screw itself is kept from mov- 
ing by Penn's method within the bearing, and the whole is 
bolted down to a base plate. The modulus for the dimensions 
is the same as before. An application of this form is shown in 
a 126, 



I 121. 

Thrust Bearings with Wooden Surfaces. 

For bearings which are operated wet, the use of Lignum 

Vita has been found to give the best results. The wood is 

inserted in a similar manner to that shown in \ 117, the pieces 

being made in the form of plugs. In Fig. 329 is shown the 

step of a screw propeller shaft of this 

type. The plugs are inserted in a 

bronze plate, and the end of the shaft 

faced with bronze. 

A bearing of this form on the 
" Orontes " had 37 plugs each i}i" 
diameter, and on 50 H. P. nominal 
English gunboats the thrust plates 
have 7 plugs each 1" diameter. Both 
these examples are bv James Watt 
& Co. 

Collar bearings with surfaces of 
wood are often made ; these should 
be always worked under water. Penn, 
to whom the introduction of such 
wooden bearing surfaces is mainly due, 
has especiall}' used them in various 
bearings in the length of a screw pro- 
peller shaft, the lower half of the shaft 
running in a water trough. The usual 
construction of the thrust ring between the hub of the screw 
propeller and the stern post is shown in Fig. 330. A is the 
shaft with a bronze sleeve fitting into the wooden lining of the 
hole through the stern tube ; B is the hub of the screw pro- 




peller ; C the thrust ring with its wooden plugs ; D is the 
nozzle on the end of the stern tube showing the stiffening ribs 
which assist in receiving the thrust. The parts B, C, D and 
E are of bronze. 




Fig. 331. 

Fig- 331 shows a form of thrust ring used on the imperial 
steamships "Kaiser," "Friedrich Karl," "Preussen," "Vineta," 
" Frej-a," "Ariadne," "Nautilus" and "Cyklop." The ring 
is made in halves, and can readily be removed and replaced. 



THE CONSTRUCTOR. 



77 



The two axial projections enter into recesses iu the flauge ou 
the end of the tube, and prevent the thrust ring from revolving. 
The dimensions of the wooden bearing surfaces on the various 
ships above named are approximately as given in the following 
table : 



jKaiser. 


Friedr. 
Karl. 


Preussen. 


Vineta. 


Freya. 


Ariad- 
ne. 


Nauti- 
lus. 


Cyklop 


b' 

Surface 
sq. ft. 


28" 

('Y^" 

4.078 


24^'/ 

IW 
0.740 


2634:" 

1%" 
2.629 


l(>yz" 

l%" 

I.ISS 


WA" 

A%" 
1.840 


I9X" 

A'A" 

1.840 


xoy^" 8K 

0.476 0.238 



In the "Wasp'' the thrust ring is made with 6 sectors of 3 1S6 
sq. ft. surface , in the " Leipzig" there are So small sectors with 
a total surface of 2.422 sq. ft. The use of such thrust rings filled 
with blocks of lignum vitte has been most successful in vessels 
of the German navy, and the wear on the wood has been so 
slight that renewal is rarely necessary. 

I 122. 
MuLTipivE Coivi,AR Bearings. 

For thrust bearings which are subjected to heavy service, the 
multiple collar bearing is most valuable. These are very gen- 
erally used to receive the thrust of screw propellers, but are 
also used in other situations, as, for example, large turbines, 
also centrifugal machines of great size and weight, such as are 
used in sugar refineries. The forms which may be given to 
these bearings are quite varied ; but in every case the most 
important consideration is the pressure to which the various 
surfaces are subjected. 

For pillow blocks in which the shaft is made with several 
collars, the boxes may be cast iu bronze with internal collars 




, Fig. 332. 

as shown in Fig. 332.* For larger dimensions, the boxes may 
be strengthened by ring shaped ribs, let into recesses in the 
cap and body of the bearing. 

Example : — The thrust bearing on the " City of Richmond,'* built by Todd 
& MacGregor, of Glasgow, from the designs of Jaffrey.f has 12 rings ; inside 
diameter, 19''; outside diameter, 23"; total length of the bearing, ^3%'^ "The 
boxes are strengthened by three ribs of K" depth by 4" wide. The engines 
indicate 3340 H. P., and the speed of the vessel is about 1343 feet per minute 

James Watt & Co. make the boxes free in the bearing, and 
support them by set screws at the ends, as shown in Fig. 333. 

On the "Medusa" and 
"Triton " four set screws 
are used in each flange, 
the shaft being 7" diam- 
eter, with six rings. In 
the "Jason," by the same 
firm, there are six set 
screws in each flange, the 
shaft being 12" diameter, 
with eight rings, j 

Boxes of cast iron lined 
with white metal are 




1^-4- 



FiG. 333- 



sometimes used by various makers, as, for example, in the 
"Mooltan" by Day & Co., in which the shaft is 13X" diameter, 
and has twelve rings. The design shown in Fig. 334, which is 
a French pattern, uses an adjustable bearing lined with white 
metal. 

In Fig. 335 is shown a form of thrust bearing in which the 
rings are made of bronze separately, and fitted to the body and 
cap. This form is the design of Ravenhill & Hodgson. Espe- 
cially to be noted is the arrangement of bolts. These are iu 
two sets, the first securing the body of the bearing to the sole 
plate, and the second being the cap bolts. The ledge or tongue 
which is let into the sole plate is arranged with a space as 
shown on the left, in which a key is fitted to provide for the 



take-up of the wear upon the rings. The cross section in upper 
right hand portion of the illustration shows the construction 
and application of the bronze rings. The arrangement provides 
for a constant distribution of grease, thus preventing the rust- 
ing of the journal by the application of water for cooling. 




Fig. 334- 

In Figs. 336 and 337 is shown a thrust bearing by Penn, as 
used on the " Kaiser." Here the bearing surfaces are made in 
separate rings of still simpler form than the preceding. These 




Fig. 335. 

rings, which are made of bronze, are in halves for convenience 
of construction. In the "Kaiser" d is equal to iSj^", and there 
are eight rings ou the shaft and in the bearing. The six bolts 




* See Armengaud, Vignole des Mecaniciens. PI. 13, Fig. 32. 
t Engineering, May, 1875, p. 403. t See Burgh. 



Fig. 336. 

are arranged so as to act both as cap bolts and fastenings for 
the bearing. The adjustment for wear is similar to the pre- 
ceding case. The dimensions are based on the same modulus 
as already given, viz. : rfj^ 1.15 i/-!- 0.4". 

A most noticeable form of thrust bearing is that of Maudslay, 



78 



THE CONSTRUCTOR. 



shown in Figs. 338 to 340, as used on the "Elizabeth." For 
each collar on the shaft there is provided a separate ring and 

support, with means for ample 
lubrication. The bearing rings 
are made of horse shoe form, 
and are of cast iron lined with 
white metal. The collars on 
the shaft dip into an oil 
trough. They are also pro- 
vided with oil cups" above, so 
that as in the case of the car 
axle box previously described, 
lubrication is supplied both 
above and below. Each ring 
may be adjusted by its own 
set screws, or all can be ad- 
justed together. The propor- 
tious are all based upon the 
previous modulus, d^=^\.i$d 
+ 0.4", and the shape and 
dimensions give an excellent 
appearance. In the " Eliza- 




\ 123- 

Examples of Thrust Bearings. 

The following examples are taken from twelve of the most 
important vessels of the German navy, the data being furnished 
to the author with the approval and authority of the Chief 
of Admiralty. The power and speed of the engines and the 
velocity of the vessel are all most important data, and are 
obtained from of&cial tests. From these may be obtained, as 



-»- -1.01 -^1 



beth' 



Fig. 337. 
the shaft is I2>'2" diameter. 




iiBlM "»J2» 




Fig. 339. 

in \ 100, the maximum pressure upon the thrust bearing sur- 
faces. It is important to observe that in only two cases out of 
the twelve was a thrust ring used between the stern post and 




-B — T V 

Jozii 



c) (|) <i) I (t) cl3r:v:>-'(|) 



Fig. 338. 



Fig. 340. 

propeller hub. The elasticity of the hull of the ship may some- 
times cause the entire force to be thrown on the thrust bearing, 
and at other times much may be taken by the thrust ring. The 
data given in the table will also be found valuable for other 
purposes. 



EXAMPLES OF THRUST BEARINGS. 



No. 


Name 

OF 


Builder 

OF 






(LI 

5 


11 


o'C 


t4-. 

.2 a 


01 . 

-^ 


u5 
. u 

§•5; 


u . 

II 




go 


tfl 

3 




Vessel 


Engines. 


^1 




£S3 


01 

2; a 


S« 




a 2 


II 

V 
(5 JO 


3 


OJ p. 

^0 


OS 


























on 




No 


I 


Armored Frigate 
Konig Wilhelm. 


Mandslay Sons & Field, 
London. 


8325 


1491 


18" 


63.86 


6 


Anti- 
mony. 


8.467 


24M" 


18" 


Bearing 

cooled 

with 

Water. 


Worked 
well. 

Ran warm 


thrust 

ring 
in stern 

post. 
Thrust 


3 


Armored Frigate 
Kaiser. 


John Penu & Sons, 
Greenwich. 


7803.3 


■457 


18" 


77.00 


s 


Brouze. 


7.104 


23" 


181V' 


Ditto. 


before the 
thrust ring 
was applied. 

Made with- 
out thrust 


nngm 
stern 
post, 
^121. 


3 


Armored Frigate 
Friedrich Karl. 


("Societe des Forges et ") 
< Chantiers de la Mediter- > 
(ranee, Marseilles. J 


35°3 


1328 


15" 


61.82 


11 = 18^8" 

i=2cy," 


White 
Metal. 


8.004 


iS/a" 


15^" 


Ditto. 


ring and ran 
warm. Since 


Ditto. 


























its applica- 






























tion, works 






Armored Frigate 


rstettiner Maschinenbau ") 






















well. 




4 


Preussen. 
Decked Corvette 


\ Aktiengesellschaft Vulkan \ 
(in Bredow bei Stettin. J 


43S6.7 


1408 


.6K" 


64.5 


8 


Bronze. 


5.371 


20/s" 


^6%" 


Ditto. 


Worked well 


Ditto. 


5 


l,eipzig. 
Decked Corvette 


Ditto 
John Fcnn & Sons, 


35193 


1437 


16" 


72.4 


8 


Bronze. 


4.816 


igys" 


16" 


Ditto. 


Ditto. 


Ditto. 


6 


Vineta. 


Greenwich. 
(■Miirkisch-Schlesische Ma-~) 


1359.3 


1 120 


loj^" 


67.9 


6 


Bronze. 


1.489 


125-3" 


io;!s" 


Ditto. 


Ditto. 


Ditto. 


7 


Decked Corvette 


-< schinenbau, und HUtten > 


2598.8 


'.=i57 


12K" 


S2.52 


8 


Bronze. 


2-528 


15" 


12I/;" 


Ditto. 


Ditto. 


Ditto. 


Freya. 


(.Aktiengesellschaft. J 






















Ran warmed 
first, after- 




8 


Decked Corvette 


Ditto. 


1726.9 


1282 


iij-s" 


80.24 


7 


Bronze. 


3.391 


14%" 


1154" 


Ditto. 


Ditto. 


Ariadne. 
























ed well. 






Decked Corvette 


Mazeline & Co., 












Anti- 












No 


9 


Augusta. 


Havre. 


1127 


1245 


11'' 


62.09 


II 


mony. 


5.177 


14A' 


iiYt" 




well. 


thrust 
ring. 
Fitted 




Gunboat 


Moller & Hollberg 


























Nautilus. 


in Grabow. 


504.2 


1047 


7K" 


109.30 


6 


mony. 


1. 159 


9K" 


7%" 


Ditto. 


Ditto. 


with 






rstettiner IMaschinenbau ) 
























II 


Cyklop. 


\ Aktiengesellschaft Vulkan \ 
(inBrcdow bei Stettin. J 


245.4 


894 


ifi" 


143.89 


4 


Lignum 
VitEe. 


0.496 


IV*" 


5^3" 


Ditto. 


Ditto. 


Ditto. 


J2 


ArmoredGuuboat 
Wespe. 


J Aktiengesellschaft, Weserl 
tin Bremen. j 


799-7 


1054 


6%." 


13S.83 


i=io%" 
8=9?'3" 


Bronze. 


1.728 


9^3" 


7J^" 


Ditto. 


Ditto. 


Ditto. 



THE CONSTRUCTOR. 



79 



CHAPTER VII. 

SUPPORTS FOR BEARINGS. 



I 124. 



Generai< Considerations. 

The function of a support for one or more beariugs is to hold 
them iu a firm and definite position with regard to the frame 
or other parts of a machine. Such supports are nearly always 
made of cast iron, and in the following treatment of the subject 
this material is the only one considered. 

Simple supports are those which are intended to hold but 
one bearing, in distinction from those supports which are ar- 
ranged to receive several. In both cases the following consid- 
erations should be observed as closely as may be, when, as is 




Fig. 341. 

usually the case, the shafts which the bearings carry are fitted 
with gear wheels which should be near the bearings. 

1. The bearings should be as near to the hubs of the gear 
wheels as practicable. 

2. The pressure upon the journal should, in no case, act in 
the direction of the joint between the boxes. 

3. The support for the boxes should be so arranged as to 
allow the easy removal of shafts and gear wheels. 




4. The number of bearing surfaces should be made as few as 
possible, and all finished surfaces should be capable of being 
finished at one setting on the planing machine. 

5. Whenever possible, and especially in situations of difficult 
access, the bearings should be so disposed that the boxes may 
be removed and renewed without involving the removal of the 
shafts from their position. 



Simple Supports. 

A simple Support for a single pillow block is shown in Fig. 341. 
It is intended ior a bearing such as is shown iu 2 107 ; hence 
the upper portion is made correspondingly narrow. The two legs 
which form the main portions are reinforced by a cross girth, 
D E. The position of the points D and E may alwa3'S be well 
placed by observing the following method : Taking the total 
height A -B as a diameter, draw from the centre E a semi-circle 
yl G B ; take the middle point of the arc AG B ^t tT/ join B G, 
and prolong it, making G H=A F ; then join H to A, and 
draw CC parallel to HA, and ^ Cis the height from the base 
to the cross girth. The dimensions of the various parts are 
dependent upon the pressui'e on the bearing, and must usually 
be governed hy the dimensions of the pillow block and by the 
judgment of the designer. In order to meet the requirements 
of Rule 5 of the preceding section, there should be under the 
pillow block a removable plate, which may be given a thick- 
ness of o.3fi',. 

Fig. 342 is a similar form of support suitable for heavier di- 
mensions. 

Fig. 343 is a support for a wall b.=,aring. This is arranged to 
be built into the wall, and forms an opening through which 
the shaft can pass, and resembling what a builder calls a bull's 
eye window. The pressure of the journal is received by the 




Fig. 343. 



bracket bearing, which is supported on the key beneath, and 
can be removed without disturbing the shaft. One point which 
should not be overlooked is the bearing plate in the wall, 
shown in tangential dotted lines below the cylinder. The di- 
mensions in the illustration are based on the modulus d^ of the 
bearing. 




Fig. 344. 



A wall bracket support is shown in Fig. 344. This is intended 
to carry a pillow block, and the T slot for the bolt heads ena- 
bles the distance of the bearing from the v,-all to be adjusted. 
This form may be used for bearings of various sizes. A simpler 
and lighter form of bracket is shown in Fig. 345. This is 
merely an arm attached to a wall and adapted for a horizontal 
shaft. 

Frequently the joint between the base of a bearing support 
and its foundation is made with cement. When this is done, 
the base is adjusted to its position, resting upon wedges,'and the 
joint being closed with clay, the liquid cement is run in ; this 



8o 



THE CONSTRUCTOR. 



■will harden iu a few days so that the wedges may be driven out 
and the bolts fully tightened. 



CiLtunrrri 



._.,<2iii_ _ 




V 



"■■v. 



Fig. 345. 



126. 



Multiple Supports for Bearings. 

" Fig. 346_represents a bridge support. The vertical shaft A B 
comes from below, as for example, from a turbine, and trans- 
mits its motion to the horizontal shaft CD. The journal pres- 
sure acts at E, at right angles to the plane of the two shafts, 




^ 



Fig. 346. 



and at F it acts in an inclined direction downward, both from 
the pressure of the gear teeth, and also because of the weight 
of the wheels and shafts. These pressures are best received at 
E, by a yoke bearing as shown in \ 113, and at F, by a bracket 
bearing, \ 1 14, supported on an adjusting key. 

Fig. 347 shows a support for a step-bearing. Here the hori- 
zontal shaft A B runs in a bracket bearing at C, and transmits 
motion to a vertical shaft which is supported at D, by a step- 




FiG. 347- 



tearing, ? 119. The latter, as the illustration partially shows, 
is carried on an adjusting key in such a manner that it can 
readily be removed from below. The bridge which carries the 
step-bearing is bolted to the box-shaped base and the nuts for 
the foundation bolts are placed inside the base. 

Another form for similar service is shown in Fig. 348. The 
shaft A C, for the large gear-wheel terminates in the support 
and is provided with a small bracket bearing at C. On account 
of the position of the wheel, this is not very accessible. The 
bearings for the vertical shaft DEE, are intended to be of the 
form described in § 1 20, a yoke bearing being fitted into a space 
cast iu the upper part of the frame at E, while an independent 




Fig. 348. 

step at F is used similar to that shown in Fig. 328. The upper 

part of the frame is made cir- 
cular in shape, so that a cast- 
iron cover may be placed over 
the pinion, as shown in the 
dotted lines. The base plate 
is held down to the stone 
foundation by four bolts ; two 
of the bolts pass through the 
columns, as shown in the illus- 
trations, and so bind the two 
plates firmly together. The 
plan view shows how the col- 
% urns are keyed into the entab- 




FiG. 349. 



lature. ^ The base of the columns are let into the base plate as 
shown in Fig. 349, and an iron cement is used. 




Fig. 350. 



THE CONSTRUCTOR. 



8£ 



In Fig. 350 is shown a support for two vertical shafts, A and 
i?, the motion being transmitted from one to the other by 
means of spur gears. The shaft A, for instance, may be that of 
a turbine wheel, and B, the main driving shaft of the mill.* 
At A there is a bracket bearing such as shown in Fig. 314, and 
at j5 a step bearing, with a removable block beneath it, so that 
the bearing may be removed or examined without removing the 
•wheel or shaft. 

Fig. 351 shows a frame for a vertical shaft A B, which trans- 
mits its motion to a horizontal shaft D E. At C is a yoke bear- 
ing and at .S a bracket-bearing. The horizontal bevel gear is 




Fig. 351- 



inclosed in the semi-circular frame, so that a cover may easily 
be adapted, as in the previous case. The removal of the vertical 
shaft is not quite so convenient in this form as in some others, 
but presents no serious difficulty. In some cases the lower part 
of the frame is entirely closed and the shaft inclosed in a sort 
of pilaster, to avoid accidents. 

For a shaft running parallel to a wall, as at A B. Fig. 352, 
and transmitting its motion to one D E, at tight angles, the 
frame shown in the illustration is suitable. The bearing for the 




Fig. 352. 



main shaft at C maj' be a pillow-block, while a bracket bearing 
is suitable at F. The distance of the pillow-block from the wall 
is adjustable (as in Fig. 344). If the gears are equal in size 
the form may be as shown in plan in Fig. 353. In this case the 
journal at Cruas in a bracket bearing. If the construction is 




Fig. 353. 



Fig. 354- 



intended to fit in the corner of a building, the frame is modi- 
fied as shown in Fig. 354; the bearings at G and Hare then 
the same. Both these forms are shown in Fig. 355 and 356 in 
pseudo-perspective. 

Very often a main overhead driving shaft is required to trans- 
mit motion both to horizontal and vertical shafts from one 
point, and the combination of Fig. 357 is an example. Here 
the frame-work is made a portion of one of the columns of the 
building and is really simple in construction ; at A should be 





Fig. 355. Fig. 356. 



used a bracket like Fig. 313 ; at E and E, wall 
Fig. 310, and at C, a step bearing like Fig. 327. 



brackets like 




Fig. 35S shows a wall frame for four bearings. A horizontal 
shaft A B, is to transmit motion to the vertical shaft C D, and. 
two horizontal shafts E and F, by means of bevel gears. At B: 

I 

\ Di 





*Such a frame is used in a spinning-mill at Chur, the frame and one-half of the 
■ iarge gear-wheel being in an archway in the large end wall of the building. 



Fig. 35S. 



is a bracket, and at C a step bracket, as in Fig. 327, while th* 
bearings at E and Eaie wall-brackets, like Fig. 310. 



82 



THE CONSTRUCTOR. 



Bj a proper choice of journal diameters and clearances the 
seats for the four beariugs may be brought into one plane, 
and the other conditions of § 124 readily complied with. 

An examination of the fundamental principles of construction 
of supports for bearings will show that all forms may be repre- 
sented by a rigid piece adapted to hold in fixed relation two or 
more revolving bodies, in such manner as to permit the applica- 
tion of the various details of construction such as boxes, caps, 
bolts, etc. It is often desirable to sketch out in the first place 
a general scheme of the construction in order that the direction 
and manner of resistances and arrangement of parts may be 
examined more readily. The frame shown in Fig. 350 is simi- 
lar to the elementary shape of Fig. 359, which resembles a sim- 
j)le connecting rod ; which indeed the base plate really is, the 



■.--.'/ -'. 





Fig. 359. 

variations being due to the especial conditions and not to any 
fundamental difference. The bridge frame, Fig. 346, is in ele- 
mentary form Fig. 360. The step supports of Fig. 347 and 34S 
may be shown in principle either in Figs. 360 or 361, since in 
these elementary schemes a bearing may be shown either by 




J^] 



Fig. 360. 




the journals or the reverse. The four-fold bearing support just 
described may be sketched in Fig. 362. 

To show how these elementary sketches may serve, the fol- 
lowing application to one of Lemielle's ventilators will indicate. 




Fig. 363. 

Here, Fig. 363, nine bearings are to be supported. Three of 
these are for the drum, which is fast to the driving crank ; it is 
carried by the two neck bearings at A and B, and the thrust 
bearing at C. The six bearings at Z), E, F, and G. H, I, are 
for the rods of the buckets ; the supports for all of these are 
then the beams A, A, , the masonry, and the cranked rod 
B, I, D,C. 

'i 127. 
Cai,cui<ations For Iron Columns. 

The calculation of the proportions of iron columns often be- 
comes necessary in machine construction, for besides serving 
merely as portions of building construction they are often com- 
bined with machine details, and also enter into the design of 
framework as supports and similar relations. Their considera- 
tion in this place is therefore appropriate. 

Iron columns are generally considered as being subjected to 
stresses of compression, and, also within certain limits, to bend- 



ing stresses ; it is therefore important to allow sufficient latitude 
in the calculations to provide for variations in the manner of 
application of the load. 

The various methods of application may be treated as indi- 
cated in the following illustrations. Fig. 364, which show the 
three Cases 11, III, and IV, of g i5. The first shows a column 



«-f- 



b 

Fig. 364. 




hinged at both ends, the second is hinged at one end, while the 

The breaking loads of the 



third is rigidly held at both ends, 
respective forms are : 

a b 



n 



/2 



47r 



c 



the columns being of prismatic form and of a height /; / being 
the moment of inertia of the cross section and M the modulus 
of elasticity of the material ; / being taken in inches. As al- 
ready stated in § 16, experiment has shown that columns whose 
ends are faced off square and true fall under Case c, even though 
not held at the ends. If, therefore, a load smaller than that in- 
dicated for Case a, be chosen for all cases, security will be as- 
sured, even should both ends of the column be jointed.* 

We may therefore take for the greatest permissible load in the 
direction of the axis : 



P^ 0.4 TT 



.JE JE 



(109) 



If d is the diameter for a solid circular cross section, we have 
for cast iron, in which E = 14,200,000. 



/7 f - 



• (no) 



-^= 2,750,000 —, rf= 0.0245 \ I ^ p 
For Wrought Iron, E = 28,400,000. This o-ives 

^= 5,500,000— , rf = 0.0206*/ / m//3 -(III) 

Example i. For a load />= 33,000 lbs., a solid cast iron column 157.5 in. 
high, the diameter rf— 0,024s Vi ^ /" —4.15'', or about 4,V'. Under 
the same conditions a wrought iron column would be 3j4" diameter. 

An inspection of the formula shows that the shorter / becomes, 
the smaller is the value of d. The cross section must, however' 
never be allowed to become so small that the limit of permis- 
sible stress shall be passed. 

In order that the stress upon the cross section shall not exceed 
8500 lbs. for either cast or wrought iron (their modulus for com- 
pression in either case being 21,300 lbs.), d should in no case be 
taken as less than 



d = 0.0122 



sf 



P 



or the load should not be greater than 
P=6397rf^ 



• (112) 



The following table for round solid cast iron posts is calculated 
from formulas (no) and (112), and gives the loads which may 
safely be put upon columns of the respective heights and diame- 
ters given. 

The quantities marked with an asterisk are calculated from 
formula (112) and are a marked reduction upon the loads other- 
wise obtained. 



T E 
■'■Drewitz has tested cast iron columns with a load equal to tt^-^^ without 

obseri'ing perceptible alteration. Frbkam's Banzeibung, V., p. 534. 



THE CONSTRUCTOR. 



83 



STRENGTH OF SOWD CAST IRON COI^UMNS. 



d 


/==Sft. 


10 ft. 


12 ft. 


14 ft. 


16 ft. 


iSft. 


1 in. 


297 


191 


132 


97 


75 


59 


i>4 


1.504 


994 


671 


493 


377 


298 


2 


4,753 


3,055 


2,122 


1,559 


1,193 


942 


^y^ 


11,600 


7,460 


5,180 


3,806 


2,914 


2,302 


3 


24,060 


15,830 


10,740 


7,892 


6,043 


4,774 


M 


44,470 


28,660 


19,900 


14,620 


11,200 


8,845 


A 


76,040 


48,890 


33,950 


24,950 


19,100 


■5,090 


4% 


121, Soo 


78,310 


54,380 


39,950 


30,590 


24,170 


5 


154,400* 


119,300 


82,890 


61,460 


47,060 


37,180 


•SK 


186,900* 


174,700 


121,400 


89,160 


68,260 


53,930 


6 


222,400* 


222,400" 


171,900 


126,250 


96,680 


76,400 



Hollow Columns. — Cast iron Columns are gener- 
erally made hollow. The dimensions in this case 
may readily be determined from the formulje for 
solid columns. 

If the external diameter is «'(,, the internal di- 
ameter (/„ and the diameter of a solid column of 
equal strength, d, we have 




Y^=(l)' 



Fig. 365. 
• • • (113) 



d. 



The ratio of internal to external diameter — !- ^ 1/, is conve- 



niently made 0.7 to 0.8. We have for : 

0.5 0.6 0.7 0.7s 0.8 0.S5 0.9 0.95 
1. 10 1. 14 1.20 1. 31 1.52 



i, = 

~^= I.0I6 1.035 



The limits of stress fall within the formula for compression 
and the above results are close approximations. It is to be ob- 



</„ = 



122 \ 



-J- 



-fi 



\ 



(114) 



J 



or the load greater than 

P=6397rfoMi— *') 
We have for : 

■ ^^ _1= 0.5 0.6 0.7 0.75 o.S 0.85 0.9 0.95 

-^p^'=lO.^J^ 0.64 0.51 0.44 0.36 0.28 0.I9 O.IO 
-=1.15 1.25 1.40 I.51 1.67 1.89 2.29 3,20 



f: 



4 



-l/)2 



-^ Example 2. The solid column of the preceding example to support a load 
of 33,000 pounds was found to be 4.15'' diameter, and for a ratio 01 diameters 
of 0.8 for a hollow column for the same load we have rfo — 1.14 X 415 = 
4.73", say 4%", and the internal diameter di = 0.8 X 4-73 = 3-7S" say 33/^", 
giving a thickness of metal of ^/2 inch. Substituting these values in (114) we 
have for the greatest safe load, P= 6^gj X (4.73)- X 0-36 = 51,530 lbs. This 
shows the dimensions obtained above to be amply strong, r/" the lualh of the 

d\ 
column are cast of uniform thickness. If the ratio —7- had been taken as 0.7 

"0 
we should have obtained, from (114) d^ = 4.44", and d^ = 3.10, giving a thick- 
ness of metal of 0.67". 

In practice it is often necessarj' to work to a given external 
diameter rf(„ in which case, for cast iron, the internal diameter 
</i, may be found from : 



dy = d^S ^ —0.000,000,36 
and the load 

P= 2, 750,000 



~d*~ 
V- 



(115) 



in which P is the difference in supporting capacity between two 
solid columns of the diameters (/„ and (Z, respectively. 

It is necessary also in this case to observe that P should not 
be greater than P = 6397 {d^- — d^) -.. 

and d, not greater than I , ,, 

/ p > ("6) 

d,= d,\ i-0-000I5;t-J 

in order that satisfactory castings may be produced. 



E.vample 3. In a barracks in Berlin are hoUow columns of 142 inches 
height, bearing loads of 37.i3o lbs. These are made of diameter rfj = 6^" 
According to (115) this should give the internal diameter : 



di = 6.1875 Wi — 0. 
According to (116) we have 
di ^ 6.1875 



,00000036 



37^l8o^X 142- 



= 5.S8" 



V 



I — 0.C0015 



37,l8° 
(6.1875)2 



= 5-71" 



This would give a thickness of metal of about ]4". The empirical thickness 
for such a column is about 'i", and the actual internal diameter was 4%". 

Example 4. A cast iron column of 1S5 inches height and gii inches outside 
diameter has to bear a load of 275,000 lbs., and was made with an internal 
diameter of 63^ inches. According to (115), for direct resistance to thrust we 
get: 

but according to (116) : 
''i = 925 



.. 275,000 X (iSs)- 
- 0.00000036 "' ^ ^J ^' = 7.92" 



(9-=5)* 



V^ 



275,000 

-0.00015 — = 6.65" 

'9-25)= 



or very near the actual dimensions. 

These examples show how important it is, to take all the 
conditions into account, in order to avoid errors, and a careful 
examination of the circumstances attending each case should 
always be considered. 

Fluted Col/ii/iiis.— The cruciform section may serve as an ex- 
ample of such columns. The thickness and 
breadth, b and li, of the ribs may be determ- 
ined by comparison with the diameter d, of an 
equivalent round solid column by making : 




k 



= 35 
16 



not be less than : 



from which the approximate thickness d, for 
any breadth h, may be obtained. In order to 
keep within safe limits the cross section should 



bli==- 



or the load more than 



17000 



I 



(US) 



/'= 17000 i/; . 



Example 5. To substitute a cruciform column for the solid one of E.k- 
ample i, we may take h = i.sd = 1.5 X 4-15 = 6 225", 
We then have from (117) . 



4 = 4.15 X 0.59 



' 4-15 N 



■ 0.72" 



\ 6.225/ 

The safe load according to (118) would be : 

P= 17,000 X 6.228 X 0.72 = 76,200 lbs. 

For a direct calculation of b and /i we may use the following : 
30 PP PI' ' 



b = ^5 =: o ooo.oco 21 

14,220,000 - 2 /;' A ' 

and hence : 



P^ 4,762,000 



bh^ 



\ ■ ("9) 



Care should be taken that the load does not exceed the limit 
given by (uS). 

Example 6 In the new building of the sugar refinery of Waghilusel, built 
in 1859-60, are columns of cruciform section. 

Those in the basernent bear a load of 264,000 lbs., and are 78.74" high, the 
ribs being 2" X i4iV'- According to (119) these posts should sustain a load of 

2 X (i4.iS75l' 
P = 4,762,000 —— '' ,„"' = 4,386,000 lbs. 
(78.74)- 
According to (118) 

P= 17,000 X 2 X 14.1875 = 482,300 lbs. 
which is much more than the actual load. 

Columns of Angle and T Iron. — These are much used in 
bridge trusses, especially in America. (See \ 87). The vertical 
posts may be considered as columns with jointed ends. Case I, 
Fig. 364, and the upper chord is in compression and may be con- 
sidered as Case III, Fig. 364. The following figures show many 
of the forms, in section, which may be used for this purpose. 



auHH 




LJ JL 

n '^v 



Fig 367. 



The first is the column of the Phoenix Bridge Works at Phoenix- 
ville, Pennsylvania. This is shown made of four segments, but 
six or more are used. This form may be strengthened by rivet- 



84 



THE CONSTRUCTOR. 



ing flat iron between the joints of the segments. The four fol- 
lowing sections are from the Keystone Bridge Works. 

The sectional distribution of material should be chosen so 
that the i equatorial i moment of inertia on both the principal 
ase= are the same (see {7). The fifth section shows a double 
T iron, in the middle in dotted lines. This is used in bridge 
chords, where two or more such shapes are sometimes intro- 
duced. The last form is a combination of four pieces of angle 
iron recently usel for pump rods in mine shafts. The resistance 
to thrust is here dependent upon the distance between the 
guides of the rod. 

Grouped Coliuiins. — It is sometimes a question whether, in 
the support of very important loads, as well as for economy of 
material, it is not best to use two or three columns instead of 
one. If we let m be the number u^ed, instead of one, we have, 
for the supposition that the columns are in compression, the re- 
lation for similar sections. 



Fi = v/;« y . 



(120) 



Tliis shows that grouped columns use vwT times as much 
material as a single column. It is also economy of material to 
use a small number of heavUy loaded columns to sustain a given 
load. 

Example 7. This subject may also be treated by the aid of the preceding 
table. ]ff we have a load of 2Sco lbs. upon a column iS feet high, the diam- 
eter for a solid round column would be 2%'', while for four columns of 2 
inches diameter we have 4 X 740 = 2960, or about the same. The cross sec- 
tions are to each other as 4 X (2)^ : : (2.75)-, or as 16 : 7 56, or y/j, : i. 

Variations in the hei=;ht of columns affect the economy of 
material, other things being equal, to a marked degree, since 
the resistance to compression varies directly as the height (/). 
It is sometimes desirable to make a column in sev- 
eral portions, when a proportional reduction in 
height can thereby be secured. The triple central 
core of the column shown in Fig. 36S, is an ex- 
ample and is a form often used by architects in 
connection with columns of brickwork.* This is 
not as effective as a single column, since the volume 
ratio is yi ^m, i. e., _;< -v/ 3 = 0.865. 

In conclusion it must be remarked that the col- 
umns which are used in machine construction are 
usually made much heavier than the preceding 
calculations indicate. This is due to the fact that 
such columns are often subjected to bending and 
tensional stresses, as well as to much vibration and 
the additional material is needed to meet these con- 
ditions. Columns of cast iron which are subjected 
to tension, as in the framing of vertical engines, 
should be made at least double the section given by (ii2'i, ('114'), 
(1161, and (ii8). The security is also made greater in the case 
of buildings, as the result in Example 6 shows. 

§ 128. 

FORilS FOR IRO:s COLUilXS. 

The columns which are used in machine construction must be 
held down to the iron base plates of the machines, or if used in 
connection with building construction are secured to foundations 
of masonry. Heavily loaded columns are often placed upon 
foundation stones with only a sheet of lead beneath, and no 
fasten'ng, but otherwise some form of anchorage must be used. 




Fig 










Fig. 



Fig. 370. 



Fig. 371. 



The illustrations show three forms of fastening. In each case 
the sole plate is placed beneath the pavement. In the first case 
a special form of sole plate is held down to the masonry by an 
anchor bolt ; in the second the flange which is cast on the 
column is bolted to the keys shown ; the third construction (by 



Eorsig'i is arranged with a short cylinder bolted to the faced sole 
plate and made so as to give a space in which melted lead may 
be poured after the column is set in its exact position. A hole 
is left in the side of the column to admit the melted metal. The 
portion of the base of the column which shows above the pave- 
ment is made to conform to the general style of the building. 
In Fig. 369 a simple moulding is used between the plinth and 
shaft; in Fig. 370 a bead is added; and in Fig. 371 a double 
moulding of more elaborate outline is used. 




PIG. 372. 

The capitals of such columns are made in many varied forms. 
Fig. 372 shows, in section and elevation, a capital arranged to 
carry a beam and also to support the base of the column of the 
floor above. A recess in the top of the column receives the 
main beam, and affords a good place for a joint. If iron beams 
are used, this recess is made proportionately narrower. The base 
of the upper column is seciu-ely bolted down as shown.* 





Fig. 373- 



Fig. 374- 



Fig. 375- 



The capitals of iron columns afford much opportunity for ef- 
fective decoration, which in many cases is neglected, although 
comparatively easy of execution. For the lower columns of 
heavj' buildings the simple cubic capital so often found in Ro- 
manesque buildings is most suitable, and a good example is 
shown in Fig. 373.+ 




Fig. 377. 



Fig. 37S 



* For example, the columns in the vestibule of the theatre at Carlsruhe. 



A somewhat lighter form is shown in Fig. 374, and for some 
situations the various Gothic capitals are suitable, Fig. 375. In 

* other forms will be lound in Brandt's Eisenkonstruktion, Berlin, 1463. 
t Shown among other places in the Osteu. Lloyd, in Trieste, and in the 
Ar.senal at Vienna. 



777^ COXSTRUCTOR. 



all three examples the pattern making and moulding is not dif- 
ficult The form most used in machine construction is shown 
in Fig. 3/5, being something between the Roman Doric and the 
Tuscan orders, and having an echinus beneath the cap plate, 
and an astragal bead around the column a short distance below- 
Bj- Tarj'ing the distance of the latter from the former the effect 
can be modified for taller or shorter columns. 

The heavier form of the Grecian Doric is unsuitable for ma- 
chine construction and is seldom used. Jlore appropriate is 
the modified Corinthian capital shown in Fig. 377. The top is 
a cornice of overhanging leaves, terminating in an astragal on 
the shaft. By omitting the ornament the same form may be re- 
tained, as showu in the right hand half of the illustration, and 
also in Fig. 34.S. The fluting of the column is by no means ob- 
jectionable, at least in Germany. The fluted capital is readily 
cast by being made in a core box. 

Fig. 378 shows a capital of Renaissance form with octagonal 
abacus, well suited for slender columns. 

The support of beams, either iron or wooden, is best accom- 
plished by the introduction of a plate between the column and 
the beam, and this may be treated simply, yet in harmony ^vith 
the style of the rest of the work. Fig. 379 shows such a sup- 
port on the cubic capital already shown, and is adapted for very 



in which the solidity and substantial character of this fomi of 
construction- is well shown. 




Fig. 3S2. 



Fir. 3S1 



hea\'3^ construction. Fig. 3S0 shows a lighter capital, in which 
the support for the beam is made of a box form ; Fig. 3S1 is a 
still lighter design. This illustration also shows the effect of a 
iigh stj-lobate or base moulding, suitable for tall slender columns. 
As shown in this example, such bases are usually made octagonal 
in section, which approaches the Gothic stj'le, but they are fre- 
quently made round. As in architecture, the columns are usu- 
ally made tapering from below upward, the upper diameter be- 
ing o.S to 0.7 that at the base. Fig. 3S2 shows a more elaborate 
form of capital and bearer. 

Columns of cruciform section, already referred to, are often 
used in the construction of industrial establishments. The}- are 
sometimes to be preferred to hollow columns, since the latter 
are often cast of such unequal thickness as to be unreliable. 
Figs. 3S3 to 385 show such a column. Fig. 3S3 is from the Rail- 
way of St. Germain. Here the flutings extend from top to bot- 
tom and the column is swelled slightlj' in the middle. The form 
shown in Fig. 3S4, from the Tobacco Factory at Strasburg, is 
more elegant in its appearance. Here a rectangular base is used 
for the lower floor, but is omitted above. The method of 
connecting the base and column, as well as the connection 
between capital, beam, and column above, is shown in Fig. 3S5 ; 





-:^ 






m 






/ 





Fig. 3S3 



Fig. 3S4- 




These examples will serve at least to show the varietv of 
forms of columns which may be used, and the manner in which 
a little oruameut may be introduced into machine construction. 

CHAPTER VIII. 
AXLES. 
I 129. 

Various Kixds of .\s;i,es. 

Axles may be considered as beams which carry revolving or 
oscillatiug loads, and hence are provided with journals at certain 
portions of their lengths ; they may be subjected to deflection 
or to compression, as in the cases of journals already discussed, 
according as the load act normal or parallel to the axis. Axles 
which are subjected only to thrust, are not often found, the far 
greater portion being those bearing deflecting loads, although, 
many are under combined stresses. 

These may be divided into two classes : those which have the 
load applied at but one place, and those in which several loads 
are borne at various points. The first are called Simple-loaded 
Axles ; the second. Multiple-loaded Axles. 

The sections of axles of cast or wrought iron may be either 
circular or varied, and this gives rise to another subdivision in 
the calculations. The methods of graphostatics are especially 
applicable to the subject of axles, and in the following pages 
both numerical and graphical solutions will be discussed. 

A. AXLES JriTH CIRCULAR SECTION. 



SiMPtE Symmetric.\i, Axles. 

Q 




Fig. 3S6. 



86 



THE CONSTRUCTOR. 



The load Q is in this case applied normal to the direction of 
the axis, midway between the two journals, upon a seat for a 
hub, as shown in Fig. 386. The portion between the hub-seat 
and journals is called the shank of the axle. The journals are 
proportioned according to the methods given in Chapter V, 
taking /'= ^ Q, and the axle then proportioned so as to give 
approximately the same strength as the journals throughout. 
Let: 

d = diameter, I = length of journals, 
e = height of shoulder or collar, 
D = diameter of middle, or hub-seat, i^=its breadth, 
D' = diameter of shank at the junction with Z), 
e' = }i [D — D') the shoulder at the latter junction, 
a = the length of shank, 
then we have : 



£y_ if a — 0.5 d 
d^f 0.5/ 



(I2I)' 



This will give the axle the same security as the journal, so 
that the approximate stress will be .S'=S5oo lbs. for wrought 
iron, and 4260 lbs. for cast iron. If a higher or lower stress is 
desired, the journal should be proportioned for the desired 
stress, and the corresponding dimensions for the axle deduced. 




The strongest form for the shank of an axle is that of a cubic 
parabola (see § 10, No. VI), and the student will find this a valu- 
able subject for investigation. In practice it is made a portion 
of a truncated coue whose larger diameter^/?', and the smaller 
diameter = (/= 2^. The value of c^ should only be made large 
enough to provide sufficient depth for a keyway. 



Non-Symmetrical Simple Axles. 

If the two shank portions of a simple loaded axle are of un- 
equal length, as in Fig. 387, the load on the two journals d^and 
d^ will be unequally divided, and we have the proportion 

a, P. 



P._ 



P., 



a 2 



P, 



(122) 



Q 'Zi + «2 ' Q «i + "2 -• 1 "1 

The hub-seat divides the axle into two parts, each of which 
may be considered as the half of a symmetrical axle, and so the 




Fig. 388. 

■whole proportioned. The value of D' is determined for each 
shank, and the greater value taken for both sides, li a-^ = a^ 
the a.xle becomes symmetrical. 

If the seat for the load Q does not lie between the two jour- 
nals, but projects, as in Fig. 389 [a^ becoming negative), the load 
is said to overhang, and the journal /? becomes a neck journal 
(see § 92). 

We have for the relations of forces : 



5. 

Q 



Q 



, + «. Px 



(123) 



^\ Pi «1 + 0-2 

The diameter of the journal di is first determined, then a 
diameter for an assumed journal d., for the loaded point, and 
then a value D for a neck journal, taking for D the greater of 
the values for £>' and D", as given for the two ends by formula 
(121), the length of journal being made /, = v /j' -j- //. 



The cubic parabola is shown drawn in the shank a.^, being the 
shape for uniform resistance in this case, and we have for the 
diameter 6, at the root of the hub-seat : 



6 



^ 



(124) 



See (2 10, Case VI, Remark). 



Exampie.— 'L^i the load Q, acting in one direction, be 14,520 pounds, a-i = 
47-25", ^2 = 23.625'', b-=i2,", material cast iron, number of revolutions, n^ 
150. 

We have /'i = 0.5 ;2= 7,260 lbs. /'3=i. 5 ^=21,780 lbs. From the table of 
ggi we have o'x^ about 3^", tfa = about 614!", also /i = 5!^", 4 = 9^". "We then 
have: 



Z> = 6.25 



(5.62; 
-6.25 ^^ 



I^Sts 

h = ^(5.625)= + (9.375)2 



10.72, say io3;i". 



3 

9-375 



10.93, say II . 
6.97, say 7". 

2 1^2. 



Graphical Calculation of Simple-Loaded Axles. 

The determination of the forces acting upon the journals may- 
be made according to the methods given in Cases I to V, of I 39. 
In a similar manner the cord polygon may be employed as in 
? 43 and I 44 to determine the statical moments of the parallel 
forces at various points, and the polygon so constructed may be 
called the surface of moments. The simple method about to 
be given will serve as a general graphical solution of the problem. 

/. The Load Acts Normal to the Axis. 




Fig. 389. 



Fig. 390. 



(a). Hub and Load between the Journals. — Draw the line A C, 
equal in length to the distance between centres of journals, and 
upon it construct any triangle ABC, whose apex lies on the 
line of the load Q. Draw A 3 normal to A C, making -4 3 = O; 
draw 3 . O parallel to B C, and 2 . O parallel to A C ; then A . 2 
^7-^,2.3 = P.,, By dropping the perpendiculars from the ends 
of the hub-seat we may divide Q into two forces O-^ and Q.^, 
shown in the force polygon by O b, parallel to B^ B, ; giving 
A b^^ Oi, b . 2^ Qy The vertical ordinate /, at any point of the 
surface of moments is proportional to the statical moment Ji/v 
at its point of intersection with the axis, as for example the or- 
dinate /j, at the base of the journal forPj. We have in any case ; 



y'= 



-^--^Mj.. d,^ = ^'-^M, 



and hence ; 



= 4 



(124} 



from which y can readily be obtained.* 

(b). The Hub-Seat between the Journals and the Load Over- 
hmig. Fig. 390. — Draw the line A C, parallel to the axis, con- 
struct a triangle with the points A, B and C on the lines of the 
directions of the forces, drop a perpendicular from the point D, 
where D d=Q, make O . i parallel to A C, and equal to CD, 
make A . 1 . t, normal to A C, also O . 3 parallel to C B, and i 
. 3 will = Q, A 1 = P^, 2 . A = P.,. The force O is decomposed 
into two forces at the ends of the hubs, and by dropping thf: 
perpendiculars, the points C, and C, are determined, and O c 
drawn parallel to C^ C,, giving the values c . 3 and i . C for the 



* If it is desired to determine a series of values of /, beginning from f^, it 
may readily be done by using a table of cube roots of numbers such as are 
given at the end of this volume; if the greatest value of r is the starting 
point, the table of cube roots of decimal numbers is useful, the space being 
divided into ten parts and the outline laid off correspondingly. 



THE CONSTRUCTOR. 



«7 



forces at C^ and C.^ respectively. The diagram shows that at a 
point within the hub-seat the stresses are reversed and the bend- 
ing moment is zero. 




Fig. 391. 



Fig. 392. 



(c). Overhung Axle ivilh Load Outside the Journals, Fig. 391 
— Construct the triangle A B C, as in the preceding case {b), and 
place D so that Dd= O, draw A . 3 normal to A C, make O ■ 
2= CD, and parallel to A C, and draw O . 3 parallel to C B, 
and we have again A . 2 = P^, ^ . A=: P.,. Divide Q into Cj 
and C.2 and make Oc parallel to Cj C,, giving c . 3 and 2 . c for 
the forces at Q and C.,. The journal at B being uniformly loaded, 
its moment surface is outlined by a parabolic curve (see ^ 42). 




Fig. 394. 



(d). Overhung Axle, with Load bettveen Journals, V\g. ^^2. — 
Construct the triangle ^ B Ca.s in case (a), divide Q into Bi 
and B,, which gives the polygon A C, B^ B., (which is equiva- 
lent to the other one A C, jS'j B.^). In the force polygon, i . 3 
= g, 2 . I = Pj, 3 . 2 ^ /^2i ^°<i by making 01 parallel to B., B^ 
we get b . 3 and i . b for the forces on B-^ B^ and B.^ B^.* 




Fig. 395. 

//. The Load Acts Lnclined to the Axis, Fig. 393. 

The construction is similar to /,, except that the force and cord 
polygons are inclined according to the direction of O. The 
vertical projections a A, and 3 . cgive the journal pressures P-^ 
and P., ; the horizontal component of O gives the axial thrust. 

Another example is given the case of a rod worked from an 
overhung arm, as in some forms of locomotive feed cumos, Fig. 
395. The directions are here periodically reversed, and the re- 
lations of the points continually changing. 

HI. The Load Acts Parallel to the Axis, Fig. 394. 

We here have two couples : one consisting of the two equal 
journal pressures, and the other of the two pressures at the ends 
of the hub-seat (see ?. 3S). Draw the lines A B^ and C B., par- 
allel to each other, and intersecting the perpendiculars dropped 



from the ends of the hub, join B^ with B.^ and the surface of 
moments is A i?, i?., C. To find the forces, prolong Q from B 
until it intersects C'b, join it to the middle of the other journal, 
make qb^Q, and drop the perpendicular q a, which is equal to- 




Fig. 396. 

P. Make Ai = P, draw i . O parallel to A C, and O . 2 paral- 
lel to B, B^, then i . 2 is the force at b^ and 2 . i that at b.,. If 
the hub should overhang, as in the case of a screw propeller, 
Fig- 396, the diagram takes the form A B C^ C,. 

I 133- 

Proof Diagr.\ms. 

In order to calculate the resistance of a given axle to bending 
it is necessary to know the section modulus at various points. 
If all the sections are circular the moduli vary as the third power 
of the diameter. Hence the various diameters are to be cubed. 




Fig. 397. 

This may readily be done graphically by the method given itt 
'i 28. In order to compare such a diagram with one of the sur- 
face of moments as just discussed, it is necessary to construct 
them to the same scale. For this purpose take the origin O of 
the two axes A' and Y, and make Oa equal to the diameter (or 
semi-diameter) of the shank of the axle, and lay off, below the 
corresponding value Ob of its ordinate t-^, draw on a I 3. semi- 
circle a c b, draw a e normal to a c, and takiug O e as unity we 
)ia.\eOh=(Odf. Make O . i=y, O . 2=j/,, &c., and draw 
the moments to the axes of ^Yand }' as i, i', I., 2, 2', II., &c., 
and we have CI, Oil, as the desired values of y^^, y^" ■ ■ ■ 
which correspond to those of the principal diagram. 

Such proof diagrams are very convenient to show what ap- 
proximations may be made, and to detect possible errors in cal- 
culation, and shows at once any deficiency in securit}', since the 
relation of the actual stresses to the desired constant stress is 
that of the ordinates of the proof diagram to those of the theo- 
retical surface of moments. This numerical series ma)' be plotted 
in a curve, called the stress curve. By combining the theoreti- 
cal diagram with the proof diagram on an exaggerated scale, as 
shown in the illustration, the unit can be chosen to a greater 
advantage. 

i 134. 

Axles Lo.\ded at Two Points. 

In an axle loaded at two points, as in Fig. 398, the end por- 
tions are called the shanks and the middle part the shaft. If 
Qi and Q., are the loads, i the length of shaft, we have for the 
journal pressures 



/", 



s+aA I + 



Qj 



P, 



s+ai( i + 






*The conditions of this case, but with very light stresses, are found in the 
spindles of the American Ring Spinning Frame. 



ft a, + s+a, ■ O, a, + s + a, ^'^^ 

If we take the diameters corresponding to these pressures as 



THE CONSTRUCTOR. 



di and </,, and also have the shanks a^ and a.,_ given, we ma}' de- 
termine next the diameters at Dy and D., at the points of appli- 
'cation of the loads O^ and _(/,. 

To find the diameters of the shaft at various points we have, 
"taking _)/ for the diameter at any point distant x from the load 
joint 5i : 



aX Pj 



(126) 



an equation which gives the shaft the outline of a double cubic 
parabola, which in practice may be replaced by two straight 
lines, giving the shape a truncated cone. 




Fig. 398. 

The two seats for hubs are formed so as to give shoulders for 
teywa}^, and have a determinate breadth b, governed by the 
piece to be carried. In many cases such axles are symmetrical 
and the two loads are equal to each other, hence (7, = <7.,, Oy = 
Q^ We then have P^ = P,= Qi— Q.^ and y = D, the shaft 
"being cylindrical. This is also the case when P, a-^^P^ a^. 



«» 



Jd ^ 



V 




Fig. 399. 



Fig. 400. 



The graphical solution of the preceding problem is as readilj' 
■made as in the case of single loads. If we draw normals to the 
axis A D, Fig. 399, corresponding to the given loads O^ and O-i, 
also draw A a, make ai^ Qi and 1.2 = O.,, choose a pole O 
and draw the raj'S Oa, O J, 2, prolong a O to its intersection 
■d with the line of the force Q^, make b c parallel to i O, cd par- 
allel to 2 O and join d with a. Draw O 3 parallel to rf (7 in the 
force polygon and we have 2 . 3 ^ Z',, and 3 a ^ Pj and abed 
the surface of moments whose vertical ordinates t may be used 
"to determine their corresponding diameters of the axle as in I, 

^ 132- 

The intersection e, of ab, and dc prolonged determines the 
position E e oi the resultant of Q-^ and Q.,. If E e is desired at 




.^-«,«*!«f«!S?^ 



Fig. 401. 



Fig. 402. 



once, as in the method given in § 40, the previous case (? 132, I) 
is applicable since the direction of the line a d can be chosen at 
^will. 

If one load acts beyond the bearings. Fig. 400, the reversal 
point in the elastic line will appear as before ; this occurs when 
the resultant of O^ and O., falls between A and D (see \ 132, I). 
"The above mentioned shearing stress is given by i . 3. 

If the resultant of O, and O., falls outside both journals. Fig. 

401, there will be no reversal, the force P^ having the same di- 
xectiou as O-^ and O.^ ; in other respects the procedure is the same 
as before. 

Finally the resultant may just equal the force at /?, as in Fig. 

402. In this case there will be no bending stress in portion A 



B, which in the previous case was quite small ; the two lines of 
the surface of moments fall together. The shank A B and the 
journal 2A. A maj- therefore be made ver}- light, unless other 
forces than those already considered act upon them. 

The decomposition of the forces acting upon the hub-seat de- 
pend upon its breadth and the treatment is similar to § 132. 
Other variations may occur in the relations of the loads and 
journals, but the preceding examples will suffice. 

Inclined Double Loaded Axles, Railway Axles, Crane 
Pillars. 
The previous methods are almost as easih- applied when the 
loads act in an inclined direction. The inclined action is caused 
b}' various conditions, and as an example we will consider rail- 
way axles. 




Fig. 403. 



Besides the vertical load O at the centre of gravity S, of the 
car. Fig. 403, there are forces due to centrifugal action and flexi- 
bility, which produce a horizontal force which Scheffler, accord- 
ing to Wohler's researches, places at o 4 O,* so that there is an 
inclined resultant 7?, acting upon the axle. Since the value 
0.4 Q was obtained by means of measurements on cars during 
long runs, it includes the action of the elevation of the outer 
rail in passing curves. This force R is also acting on the wheel 
flanges at K^ and. A'', as well as at the journals A and D. It 
must be noted that the wheel K., opposed to H can oulj' resist 
forces acting normal to the coned face so that the angle L K.^ S' 
should be made = 90°. 

The points of intersection B and C of the wheel forces on the 
axle give the positions for the verticals (9i, O^, and the horizon- 
tal pressure may be neglected in determining the axle loads, 
these being P^ and P.,. From these the journal diameters are 
found and the greater taken for both. 

Then from the point of application E of the resultant R let 
fall a perpendicular E e and draw the triangle a d e, prolong the 
directions of O^ and 0-, to b and c, and join b and c by a straight 
line. Then drop perpendiculars from B' B" , C C", to b' b", 
c' c" and join these latter, and a b' b" c' c" rf is the cord poly- 
gon for the given conditions. The ordinate t serves to deter- 
mine the diameter J' for any journal diameter (/[, and the ordinate 
I., gives the root of the journal 

The direction K^ B is readily determined as follows : Choose 
any point on the line of R, as for example, E, join it with A'j 
and A'; and decompose R = E r. Fig. 404, along the directions 
iTATi and .£■ A', into E k,s.nA. k..i' = Ek^, draw k.^l, horizontal 
and .£■/ parallel to the given direction A', 6"' ; then /iT is the 
force at A',, and r I that at A',, whose direction is sought, while 
E k, and k.JL are the inner forces at the corner K„ of the cord 
polygon E A', A'j, and in equilibrium with the force of the known 
direction A', S' ■ 

Since the horizontal force //acts either to the right or left, 
the larger side of the polygon ass' b" b' must be used for both 
halves of the axle, as shown in the dotted lines. The cord poly- 
gon for the direct vertical load should also be drawn, and if it 
gives a greater ordinate for the shaft than 5 s', it should be used, 
the diameter of the shaft generally being smallest in the middle. 

Axles of railway cars make from 250 to 300 revolutions per 
minute. For wrought iron the journals are generallj' made two 
diameters in length. In passing around curves these journals 
are subjected to considerable endlong pressure. The shoulder 
c is generally made= \ d to \d, and heavier than usual in ordi- 
nar}' cases. 



^ Ad. Scheffler, Railway Axles. Braunschweig. 



777^ CONSTRUCTOR. 



89 



In many countries standard proportions for axles have been 
adopted. Those of the Prussian railways are as follows, the di- 
mensions depending upon the value of Q, which is the total 
load on each axle.* 



= 3800 kilogrammes. 


B = 


= 100 mm. 


d = 


- 65 mm 


•= 5500 


" 


115 " 




75 " 


' =: SOOO " 


(( 


130 " 


11 


85 •' 


' = loooo " 


" 


140 " 




95 " 



The journal length /varies between i|:/ to 2)^ d, according to 
judgment. These proportions are for wrought iron ; if steel is 
used Q may be increased by 20 per cent. For iron axles the 




Fig. 403. 

pressure upon any one journal should not exceed =3 Q. These 
figures give a stress of 6.4 to 8.3 kilogrammes per square milli- 
metre or 9000 to 12000 lbs., and the pressure/ from 0.30 to 0.41 
kilo, or 326 to 593 lbs. 

In Fig. 405 is shown a steel axle for the Royal Eastern Rail- 
way, with its wheels, all dimensions being in millimetres. 

In England a standard axle has been adopted as shown in 
Fig. 4o5,t and the standard American axle is similar .J The 




FlG^ 406. 

value of O in this case is about 22000 lbs. In France there has 
been no general standard adopted, but the various roads have 
adopted forms — for regular use. The Paris-L3'0ns-Mediterranean 
Railwa}' has eight forms. The form Is'o. S has (/^ 85 mm. (Sj's"), 
/= 170 mm. (6,54 "), length between centres of journals^ 1925 
mm. (7514"), diameter of hub-seat ^= 125 mm. (4xf ), diameter 
of the axle in the middle = 105 mm. {4}i"). 




Fig. 407. 

Crane pillars maybe considered as axles subjected to inclined 
stresses, as the following example will show. The crane shown 
in Fig. 407 is subject to the load /., and also its own weight G, 
and the resultant of these is at O (see examples in ^ 34). 

At Jl and B are bearings, and the pillar is held in a base plate 
at CD, the plate being secured at £ F. In order to determine 
the forces at E and/^ construct the cordpolygont'y^i^, and force 
polygon f 2 I (9, in which 2 . \ = O, i . e= O^ the i^orce at F, 



* These dimensions are given in the metric system as representing Conti- 
nental practice. 

t See Engineer, Nov., 1S73. 

tSee Engineer, June, 1S73. The M. C. B. standard varies slightly from the 
above (Trans.) 



e . 2 the force O, at E. All three external loads act parallel to 
the axis, so that we can use the method shown in Fig. 394. In 
the diagram to the right we make q-^ 172= O, and q., q.^ parallel to 
y? ^1 normal to A B. These lines then represent the horizontal 
forces /'j and P^ at A and B. 

The bearing at A carries the entire vertical load, and hence 
we have at A the inclined resultant P/ of Q and /",. We now 
draw C_/"„ normal to A C,f,f^ ^= O, draw y, D and also /if^ 
parallel to C/i, thenf^/^ will give the magnitude of a force act- 
ing right at I) and left at C- In a similar manner, draw e^ i?2 = 
Q.,, and draw <?, D, and make e., e^ parallel to ^, C, and c, e^ will 
be the magnitude of a force acting left at C and right at £). 

We therefore have P^ ^J\/.^ -\- t\ e., and Pi = e^ e^ -{-/^f.,- The 
vertical pressure of the pillar itself is all taken at £>, hence we 
get for its vertical component O =fif\ — e^ e,, which combined 
with Pi gives the resultant P.^. This is proved by the intersection 
of P/ and /■/ at S must fall on the line of the resultant of P^ 
and P,. 

If we neglect the compression in the direction of the axis, 
we may now draw the force polygon a 2 3 O of the forces P^, 
P.,, P3, P„ as shown at the left of Fig. 407, and thus obtain the 
surface of moments abed. 




Fig. 40S. 

A crane with swivel column, to which the jib or boom is 
rigidly attached, may be examined as shown in Fig. 40S. The 
position oi O = L -\- G \s taken as before, making q-^ q.^ repre- 
sent Q, draw A q^ normal to the axis, join 9, D and draw ^2 ?3 
parallel to A q^ till it intersects with q^ D. We then have q^ q^ 
for the horizontal force P^ at A, and q., q^ the corresponding 
horizontal P^ at D. 

The step bearing at D will be subjected to an inclined thrust, 
the resultant of O and /'j. 

In a similar manner we obtain the horizontal forces P.^ and P^ 
equal and opposite, and acting at B and C, and the resultant of 
the force at B with O gives the inclined force due to the rod B 
E. The four horizontal forces have the same action as the load 
on the axle in Fig. 394. We may thus obtain the surface of 
moments abed, which shows a zero point for bending moments 
between B and D , and also indicates a forward bending above 
and a backward below. In the force polj'gon 2 a-= P.,, (Z 2 = P^, 
21 = Pi and I 2 = /"j. 



V \ ■ 

1a. bI ic yP; 




Fig. 409. 

§136. 

Axi,ES WITH Three or more Bearings. 

The number of bearings for an axle is often as great as four. 
In such a case the forces and moments ma)' be found as follows : 



90 



THE CONSTRUCTOR. 



Starting at a, Pig. 409,- with the given forces I to 5, we form 
the force polygon a^ O, and, according to J 40, the link poly- 
gon abed efg, and join the closing line g: a, parallel to O 6, 
in the force polygon ; giving 5 . 6 = the force P.^ at G,6 . a~ 
the force F^ at A. From P^ and P^ the journals d-^ and d.^ may 
be determined, and the ordinates of the cord polygon give the 
means of obtaining the axle diameter as before. 
^ The intersection^, oi ab unAfe, prolonged, is a point of the 
line of direction G g, of the resultant of the forces i to 4. If it 
is desired to find the successive resultants of the various forces 
as they are combined (see I 40), it will be found convenient to 
choose O, so that a/ will be parallel to A F. The inclined link 
polygon may also be transferred to a closing line parallel to 
A F. 

If the shanks of the axle overhang the journals, as in Fig. 
410, the procedure is similar to the preceding. Beginning at 
the point a, the force polygon « 5 C> is coustructed, and the first 
side of the cord polygon b a, drawn to the line of the first force, 
the second to the line C c, of the second force, and so on to the 
closing line eb. The first and nth line of the cord polygon in- 
tersect as before on the line H h of the resultant. Variations 
on these examples may occur, as when the loads act in inclined 
directions, or opposed to each other, the methods being similar 
in all cases. 

I 137. 

Axi,ES WITH Inclined Loads. 

The analytical investigation of axles becomes more difficult 
when, as in Fig. 411, the loads act in different planes, but the 
graphical method is readily applied. The force polygons A O^ 




Fig. 411. 



I, and D O., 2, are constructed for the forces Qi and O.,, re- 
spectively. Fig. 412, the polar distances G O^ and HO., being 
made equal to each other, so that the closing lines of the two 
cord polyjjous A b' £>, aud A c" D, coincide in A D. Then 
construct the second cord polygon with the inclined ordinates 
B B" = Bb", CC"= Cc", &c., making the angle u with the 
force plane of the ordinates of the first polygon, and inclined 
backwards as drawn. Then make B b = B" b\ Cc= C" c' , 
E e ■= E" e' , &c., and draw the cord polygon A befcD, fitem 
which can be obtained (according to \ 44) the bending moments 




Fig. 412. 



for the axle. The line b efc is a curve (hyperbola), A b and 
c D are straight lines. Draw O^ O/ parallel to A i. On O./ par- 
allel to £> 2, and drop the perpendiculars O/ J and 0/ /\, and 
A I will be the force on the journal /\, and D A^that at P.,, 
measured on the scale of the force polygon. Th'eir directions 
are determined by combining A G with H :!., and D II viith. G 
1 at the angle u. 



B. AXLES WITH COMBINED SECTION. 
I 138. 

Annoi^ar Section. 

If it is desired to make au axle with annular section, or iiE 
other words, a tubular axle, the journals should first be calcu- 
lated, according to the method given in ^ 90 for tubular journals, 
and then, retaining the same proportional thickness, determine 
the dimensions of the other parts in the same manner as for 
solid axles. The most commonly used ratio of internal to ex- 
ternal diameters is o . 5. Instead of doing this, all the dimen- 
sions for a solid axle may be determined, and then having chosen 
a ratio for diameters, increase all the sizes according to formula 
(95). See also I 141. 

?I39- 
Axi,ES WITH Cruciform Section. 

In cases where axles are made of cast iron the cruciform sec- 
tion, with circular centre and four ribs, is sometimes used. The- 
shanks are then usually made of the ordinary conoidal form. 
Fig. 413, and in some cases the ribs gradually swell into a junc- 
tion at the ends with the central core, Fig. 414. 






Fig. 413-414. 

In designing such an axle, first proceed as if drawing a solid 
■ circular section as shown by the dotted lines, of the diameter 
corresponding to the 'portion A'' when the ribs join the head. 
Then for any point {x) of the shaft : 

y = the diameter of the assumed round axle, or equivalent 

conoid , 
h = height of ribs ; 
b = thickness of ribs ; 
k = diameter of core ; 

and the proportions are obtained from the following formula : 



+ 



16^ 



h J^ h 



(-1)1 

(I27> 



This formula serves for the pure cruciform section, without 
core by making k = b. 

The results vary so slightly when k = 0.2 h, that the follow- 
ing table may be used for both sections : 



b 

h 


h k 

Values of — when — 

y k 


0.80 


0-75 


0.70 


0.65 


o.bo 


0-55 


0.50 


0.45 


0.40 


0.35 


0.30 


0.25 


0.20 


0.05 


1.30 


1.40 


1.50 


1.61 


1.72 


1:84 


1.94 


2.04 


2.1.S 


2.18 


2.22 


2.26 


2.27 


0.06 


1.30 


I -39 


1.48 


t.s8 


1.68 


1.79 


1.87 


I -95 


2.02 


2.07 


2. II 


2.13 


2.14 


0.07 


1.29 


1.38 


1.46 


1.56 


1.6.S 


1.74 


1 82 


1.89 


1.94 


1.98 


2.00 


2.02 


2.02 


0.0S 


1.28 


1.36 


1.4=; 


I.S3 


1.62 


1.70 


1.76 


I.«3 


1.87 


I.ql 


1-93 


'•93 


1.93 


0.09 


1.27 


1-35 


1.43 


i-.Si 


I ■,=59 


1.66 


1.72 


1.77 


1.81 


1.84 


1.86 


1.87 


1.87 


o.io 


1.27 


'•.34 


1.42 


1.49 


l.=;6 


1.S3 


1.68 


1.72 


1-75 


1.78 


i.8o 


I. So 


1.81 


0.1 1 


1.25 


l-,33 


1.40 


1-47 


1.54 


1.60 


1.64 


1.68 


1.71 


1-73 


1.74 


'•75 


'•75 


0.7 2 


t.2S 


1.32 


1.39 


1-45 


LSI 


I..S7 


1.61 


1.64 


1.67 


1.68 


1.69 


1.70 


1.70 


0.13 


l-s.'^ 


1.3 1 


l-.3« 


1-43 


1.49 


I..S4 


I.,S« 


1.61 


1.63 


1.64 


'.65 


i.fs 


1.65 


0.14 


1.24 


1.30 


1.36 


1.42 


1.47 


I.";! 


i-SS 


1-57 


i-,S9 


1.60 


1.61 


1.61 


1.61 


0.15 


1.23 


1.29 


I.3.5 


1.40 


1-4.'; 


1.48 


1.52 


'•54 


1.56 


'■57 


1.5s 


1.5s 


1.58 


0.16 


1-23 


1.2S 


I -.34 


'.3« 


1-43 


1.46 


1.49 


r.52 


1-53 


1.54 


i-SS 


1^55 


'•55 


0.17 


1.22 


1.27 


1-33 


1-37 


1. 41 


1.45 


'•47 


1.49 


1-50 


i.S' 


1-52 


'•52 


1.52 



THE CONSTRUCTOR. 



91 



Exajnple i.— Simple Cruciform Section. — If the height of the ribs at any 
point is made double the diameter y\ of the ideal conoid, we have in the 
third line of the table, lirst and last columns, the thickness of rib b = 0.07 h. 

Example 1 — Suppose a core to be used and at any given place h = i.^y, 
and k = 0.6 /:, we have, according to line S, columns 6 and i, 6= 0.12 of the 
height h at the same place. 




? 141. 



Compound Axi,es for Water Whe;e;i,s. 

In Fig. 417 is chown an axle for a water-wheel, made of cast 
and wrought iron. This was made to replace a broken axle of 
wrought iron, for a wheel 32. S feet (10 m.) diameter, 19. 6S feet 
(6 m.J in width.* The load is carried at four points, as shown 




Fig. 415- 



Fig. 416. 

giving a total of 82,104 Ihs.f The shaft consists of a drum of 
sheet iron yi" thick and 44" outside diameter, made in three 
sections riveted to the central spiders of the wheel. The two 
journals are fitted to the cast iron heads with a slight taper, the 
ends being prolonged into the middle of the drum, where they 
are drawn together by a right and left hand nut. The journals 



We may make b, constant and determine k, or let k be con- 
stant and d vary. The latter case is shown in Fig. 415. Here 
the shanks are also cruciform in section, and the hub-seats are 
made to receive keys, as shown in both sections, and the central 
one is strengthened by transverse ribs, A small auxiliary jour- 
nal is shown at the end of the main journal, and is very useful 
in erection. 

I 140. 

Modified Ribbed Axle. 

For heavily loaded axles the form shown in Fig. 416 is suit- 
able, the ribs being provided with flanges along the edge. Fair- 
bairn has used such axles for water-wheels, and Rieter& Co., of 
Winterthur have made them for the same purpose. The pro- 
portions are determined by taking the diameter y, of an ideal 
shaft of circular section, and calculating /;, as before. We may 
then make the flange thickness c = b, the thickness of the ribs, 
and then the flange breadth b-^ is obtained from the formula : 



b 



1 + 



\(>\ hi h \ h I 



(128) 



from which the following table has been calculated : 



b 

A 

0.05 
0.06 


Value of -'-, when — 
b y 


l.IO 


1.20 


1.30 


1.40 


1.50 


1.60 


1.70 


1.80 


1.90 


2.00 , 


















3-64 


2-75 












7-94 


6.17 


4.81 










6.99 


5-38 


413 


317 


2.34 


1.07 


0.07 








6.70 


5-12 


391 


3-45 


2.24 


1.61 


I.OI 


0.08 
0.09 

O.IO 






6.82 


516 


3-91 


3-45 


2.24 


1.61 


1. 17 









5-45 


4.1 1 


3.10 


2-33 


1-73 


I.OI 






6.00 


4.48 


3-37 


2-53 


1.89 


1-39 








0.1 1 




5-05 


3-77 


2.82 


2.1 1 


1-57 


IIS 








0.12 


6.56 


4-34 


3-23 


2.42 


1.80 


1-34 










0.13 


5-73 


3-78 


2.S1 


2.10 


1..56 


1. 15 











0.14 


5.06 


3-34 


2.4S 


1.85 


1.38 


1. 01 









The ratio between b-^ and b is never made greater than 6 to 7, 
and as it does not fall below unity the table is only given be- 
tween these limits. The profile is determined for a few points 
and these are joined by a continuous line. 




C D 

11,924. 11,924. 



j -~Lf- -"-'^ 




Fig. 417. 

are ^Ys," diameter and 1 1 " long. The circumferential joints in 
the drum are strengthened by pieces of angle iron as shown. 
The stress in the shell of the drum is only 3100 lbs., and on the 
riveting about 6400 lbs. 

I 142. 
Construction of Rib Profiles. 

In drawing the curved outline of ribs such as shown in the 
preceding designs, the following methods may be employed. In 
the various diagrams A B is the geometric axis of the piece, ,S 
the highest point of the curve, and /C the lowest point, these 
both being already determined. 

I. Circular Arc. — This can only be used to advantage when 
on such a small scale that it can be drawn with compasses or 
trammel. 




Fig. 418. 

2. Parabola.— V>rs.-^ S D and C K parallel to A B, divide 5 
D into any number of equal parts, as for example, six parts, 
and divide Z) A' into the same number. Drop perpendiculars 
from I, II, III, &c., join the lines 5' i, 5 2, 5 3, &c., and the in- 
tersections of these with the perpendiculars I, II, III, &c., will 
be points in the parabola. 

3. Sinoide. — Draw 5 Z> and (TA'^parallel to A B ; with a radius 
A S draw a circle about A ; divide the arc 5 £, cut off between 
S D and (Tiifinto six, or any number of parts ; draw from the 
points of division, lines parallel to A B, and from I, II, III, &c., 
perpendiculars to A B, and the intersections will give points in 
the sinoide. 



*This wheel belongs to the Societe des Faux du Rhone, at Geneva. 
Annales du Genie Civil, 1866 and 1S72. 
t See diagram in Fig. 409, where the loads are in this proportion. 



See 



92 



THE CONSTRUCTOR. 



4. Elastic Line. — By bending an elastic rod of uniform pris- 
matic cross section, keeping it upon the points K^, S, and A'j, 
the elastic curve may be drawn directly from the rod, using it 
as a ruler. For large sizes the rod may be ;54" to iX"thick, and 
kept under water : for smaller sizes, about ){'' thick is sufficient. 



Let: 




&- 



5. Cardiode. — The following method may be used for drawing 
the curve directly in the pattern loft. A wooden template S' K 
E C (Fi.g. 420) is made, in which E C and E S' are straight 
edges, and C S"^CS, and CE^C K. Guide points are 
placed at C and A', and the edge C E kept against the point C, 
and the edge S' E against the point K. The point S' of the 




Fig. 420. 

template will then describe a cardiode curve and by attaching a 
pencil point at S' it may be drawn directly for pattern makers' 
use. 

The most convenient method in practice is to obtain a few 
points by (2) or (3), and then join them by a flexible spline or 
ruler. 

? 143- 
Wooden Axi,es. 

For some water-wheels axles of oak are still used, and these 
are made polygonal in section. They are made prismatic, the 
diameter being at all points equal to that necessary at the point 
of greatest stress, and the methods of attaching journals are 
shown in i< 102. The diameter may be obtained by multiplying 
the diameter for cast iron by 1.55 (the cube root of the ratio of 
the modulus of cast iron and oak). 

This must be the full actual diameter, as it is sometimes 
weakened by mortises cut for the arms of the wheel. Should 
this give a less diameter than required for the attachment of the 
journals, the diameter at the latter point must be taken for the 
whole axle. The choice between iron and wooden axles must 
be governed entirely by local reasons of cost and convenience. 

Example. — A. water-wheel axle with shanks 106.25" long is loaded so that 
the journals of cast iron require to be 3^" diameter and 55^" long. Accord- 
ing to the formula in ^ 130 we have 



D ■- 



2.625 



For corresponding strength in wood, the axle should be at a minimum: 
D' =|i2 X 1.55 = 1S.6". 



CHAPTER IX. 

SHAFTING. 

I 144. 

Calculations for Cylindrical Shafting. 

In machine construction those axles which are used to trans- 
mit twisting moments are called shafting. 

In order to fulfill this purpose two requirements must be met : 
first, the ultimate strength must be sufficiently great, and se- 
cond, the torsional spring must be kept within proper limits. 
In actual practice, shafting is subjected not only to torsional 
stresses, but also to bending due to the weight and pressure of 
gears, pulleys, levers, etc., which are carried. These latter in- 
fluences will not be considered at first, and the calculations 
made only for round, solid wrought- and cast-iron shafting. 



P^ the force acting to rotate the shaft ; 
R = the lever arm at which it acts ; 
7V= tiie horse power transmitted ; 

n = the number of revolutions per minute; 

rf^the diameter of the shaft; 
L = the length of shaft in feet ; 

iJ = the angle of torsion in degrees ; 

6'^ the fibre stress at the circumference; 
G = the modulus of torsion of the material = -| of the modu- 
lus of elasticity. 



We then have for strength : 



xf: 



^\^PR. 



(129) 



and for stiffness : 



d=^. 



J2_ 

ff G 






• ('3°) 



In order to have the same security for the shafting as already 
given to journals the value of 6' should be only i the fibre stress 
(see 2 5), but in actual practice the stress is taken the same as 
for journals, viz.: for wrought iron 5=8500 lbs., and for cast 
iron 6' = 4250 lbs. 

This gives the following results for strength : 

For wrought iron shafts. 



rf= 0.091 ^^ Pi? = 3.33 ^ 



N 



For cast iron shafts, 



a' = o.ii4 ■^■^ Pi? = 4. 20 



<f 



N 



(131) 



(132) 



In taking the torsion of shafting into consideration the great- 
est allowable twist in degrees should not be over 0.075° P^r foot 
in length of shafting, that is >9°^ 0.075 ^, w'hich gives for stiff'- 
ness against torsion : 



N 



N 



and for cast iron .shafts : 

d = <:>.2S7<^~P^=5- 

jV 
The quotient of effect — is obtained from the relation to the 
n 

statical moment /■ J? as follows : 
33000 X 12 



^<1 



(133) 



(134) 



PR=- 



=: 63.020 

11 II 



(135) 



From these formulae the following table for round wrought 
iron shafts has been calculated. An inspection of the table will 
show that it is quite possible for a shaft to be strong enough to 
resist permanent deformation and yet be so light as to be liable 
to spring under its load. For example, a shaft 26 feet long, with 
a twisting force of 220 lbs. applied at one end, and acting with a 
lever arm of 20", gives a turning momeut /'A'^4400 inch lbs., 
which wottld require a shaft only li inches diameter (see column 
2). This, however, would permit far too much torsion, and in 
order that the angular deflection should not exceed the limit of 
0.075° per foot, a corresponding value oi P J? \n column 4, must 
be found, and against it in column i will be given the diameter, 
in this case about 2f " ; which, by comparison with column 2, 
gives about five-fold strength. 

For short shafts this examination of angular deflection is un- 
necessary, as for example, in the short lengths between two 
gear wheels, for here the value of 1* will be small enough in any 
ca.'^e. With longer shafts, and in all special constructions, it is 
important to consider the angular deflection and keep it within 
the given limit. 

For shafting of cast iron the same table may be used by tak- 
ing double the values for P P, or for — 

For steel shafts, whose modulus of resistance is f greater than 
wrought iron, the diameters in both cases may be taken as 

V 0.6, that is, 0.S4 times that of correspondingh- loaded wrought 
iron shafts. 

Shafting which is subjected to sudden and violent shocks, as 
in rolling mills, etc., must be made much stronger than the pre- 
ceding formulas require, and these must be classed with the 
special cases which occur in every branch of construction. 

Example I. a crane chain carrying a load of 5940 lbs. is operated by a 
drum 01 7.3 inches radius, measured to the centre of the chain; required the 



777^ CONSTRUCTOR. 



93 



diameter of the shnft to resist fhe torsion thus brought upon it. Here PR = 
43,362, which by reiereiice to columu 2 gives a diameter between^ 3 and 3j^ 
inches, sa}' sj's". This would require to be somewhat increased for bending 
stress, for which see \ 150. 

Example 2. A turbine delivers 92 horse power to a wrought iron shaft, 
making 114 revohitions per minute, and of a length = SJ^ leet. In this case 

N 

= 0.S07, which, by column 2 in the table, would require about 35^ inches 

11 
diameter. If the deflection is not to exceed 0.075° per foot, we have, in col- 

N 
unin 4, a value of ~ = 0.S03, which gives a diameter of 4j^", and with this 

n 
diameter the angle of torsion would be 8.5 X 0.075 = 0.65°. ■^ similar case in 
practice has a shaft diameter of 5^", which gives a still smaller angular de- 
flection. 

?. 145- 

Wrought Iron Shafting. 





FOR STRENGTH. 


FOR STIFFNESS (Torsional) 


d 


PR 


N 
11 


PR 


.'V 
n 


I 


1,327 


0.021 


123 


0.0019 


iX 


2,591 


0.052 


301 


0.0048 


^y^ 


4,479 


0.071 


625 


0.0099 


t-Ya 


7,112 


0.114 


1,357 


0.0183 


2 


10,616 


0.168 


1,975 


0.0313 


^% 


15. 115 


0.239 


3,t64 


0.0502 


2K 


20,730 


0.329 


4,822 


0.0765 


2^ 


27,600 


0.433 


7,061 


0.1120 


3 


35.830 


0.56S 


10,000 


0.1587 


1% 


56,890 


0.902 


18,520 


0.2941 


4 , 


84,93° 


1.347 


31,600 


0.5015 


A% 


120,900 


1.919 


50,620 


0.8032 


5 


165,800 


2.632 


77,160 


1.2240 


sy 


220,800 


3-503 


ir,ooo 


1 .7920 


6 


286,600 


4-548 


160,000 


2.5390 


6X 


364,400 


5-784 


220,300 


3.4960 


7 


455,200 


7.222 


296,400 


4.7040 


VA 


559, Soo 


8.883 


390, 600 


6. 2000 


8 


679,400 


10.780 


505,700 


8.0240 


' 8K 


815,000 


12.930 


644,400 


10.2300 


9 


967,400 


15.350 


810,000 


12.8600 


9K 


1,138,000 


18.050 


982,700 


15.6000 


10 


1,327,000 


21.050 


1,230,000 


19.5900 


loK 


1,536,000 


24.380 


1,501,000 


23.8100 


II 


1,766,000 


28.020 


1, 808,000 


2S.6800 


11;^ 


2,0lS,000 


32.020 


2,159,000 


34.2600 


12 


2,293,000 


36.390 


2,560,000 


40. 6200 



1 146. 
Line Shafting. 

In the previous discussion we have assumed that the bending 
forces upon shafting might be neglected. As a matter of fact, 
this is rarely the case, only occurring when the turning moments 
are those due to a simple force couple. Nearly all the shafting 
used for power transmission is subjected to bending stresses due 
to belt pull, pressure of gear teeth, weight of gears and pulleys, 
and to take all of these into consideration would make a very 
complicated calculation. 

In most cases ample strength will be given by taking the 
diameters according to the formulas (133) or (134). As already 
shown, these give ample strength, so that any ordinary bending 
stresses are provided for. These give reduced diameters for the 
higher speeds, shafting for high speed machinery running at 
120, 140 or even 200 revolutions per minute. 

First movers run a lower speed and are proportionally heavier, 
and the line shafting generally is gradually reduced in diameter 
in the successive ascending floods of a building. Such line shaft- 
ing is only occasionally made of cast iron, when moderate power 
is to be transmitted. 

The practice in the proportion of shaft diameters is not alto- 
gether consistent. In many cases very high stresses are per- 
mitted, as in the case of locomotives, in which stresses of 12,000 
to 15,000 lbs. are borne by wrought iron cranked axles; shafts 
of screw propeller engines usually carry 7,000 to 8,500 lbs., while 
in many instances the stresses upon line shafting are very light, 
when the high rotative speed is taken into consideration. This 
is particularly the case in England, the shafting running at 
higher speeds with a proportional reduction in diameter. The 
greatest difficulty to be encountered lies in the fact that the 
forces are rarely given with sufScient accurac}', the so-called 
"nominal" horse power which a shaft is supposed to transmit 



bearing no definite relation to the actual power. In most cases, 
however, the use of the formulas above given for stiffness, with 
a slight increase for very long shafts, will give satisfactory re- 
sults. 

A few examples will serve to illustrate the manner in which 
the methods given may be applied, and the remarks which have 
been made should be borne in mind in connection with the ap- 
plication. 

Example i. The screw shaft of a large war ship is driven by two cylinders, 
each exerting a total pressure of 176,000 pounds, on cranks of 21.75" radius, 
situated at right angles to each other The shaft is of wrought Iron, and 
between the crank shaft and the propeller it is 72 feet long, by 15" diameter. 
Calculating this for strength by forniula (131) we Jiave ; 



PR. 



= 2\/ 0.5 X 176,000 X 21.75 = 5,412,000 



: 15.98", say 16" 



If it is desired that the torsional deflection shall not exceed 0.075^ per foot 
of length, formula (a33) must be used, giving: 

' (f =0.3 i^5, 412, 000= 14.47". 

This is somewhat less than the previous dimensions, and consequently 
the deflection will be less than 72 X .075 = 5.4°. 

£.ra!J/ph' 2. In the mills at Saltaire there is a cast iron driving shaft mak- 
ing 92 revolutions per minute, and transmitting 300 horse power, the diame- 
ter being 10 inches. According to formula (134) the diameter would be : 



d = S-63 



</ 



92 



^7-56", 



so that the actual shaft is ^ stronger, and the other shafts in the mill are 
proportionally heavy. 

Exaiiiple 3. In the rolling mill at Rheinfall is a line of wrought iron shaft- 
ing, 223 feet long, transmitting 120 horse power. The speed is 95 revolutions, 

giving the ratio — = — = 1.263, The diameter for strength, as given 

« 95 

from column 3 oi the table, would be about 3^''. The actual sizes are 3%" 
in the journals, and 4" in the body. The corresponding fibre stresses are 
7,^7 lbs. in the journals, and 6,541 lbs. in the body of the shaft. According 
to the formula of Fairbairn, who designed the mills at Saltaire, this shaft 
'vvould have been made 



^ n 



or nearly eight times stronger than was actually used. 

Example ^. In the spinning mill at Logelbach there is a cast iron shaft 
^%" diameter, luaking 27 revolutions per minute and transmitting 140 horse 

power by actual measurement The ratio — = — = 5.19. 

n i-j 
Taking the double value in the table, since the material is cast iron, we 
find in column 5, that tf — 8^X. The diameter, for strength only, would be 
found by column 3 to be about •]%". 
Exaviple 5. In the same mill is a line of cast iron shafting, 84 feet long, 

N 

transmitting 270 horse power, and making 50 revolutions, hence = 5.4. 

;; 
The journals are 6^^" diameter, and the body of the shaft is the section 
shown in Fig. 413, and its section approximates to that of a cylindrical shaft 
of ZYz" diameter. For such conditions the table gives in column 3, taking 

pj 
double the value of — , we get ^=8". The diameter 6%" in the journals 

n 
gives a fibre stress of about 5,200 lbs. From the length of the shaft it is ad- 
visable to take the diameter for stiffness, which we get from the value cor- 
N 

responding to 2 = lo.S in column 5, which gives d = 8^g", which is quite 

?/ 
close to the actual dimensions. 

? 147- 

Determination of the Angi,e OF Torsion. 

In a cylindrical shaft of a diameter d, which transmits a stati- 
cal moment PR, throughout its length L, the modulus of tor- 
sion of the material being C, we have from No. I, \ 14, the angle 
of torsion. 

P R L S L 



* = - 



1?°: 



JPG _G 
-Ao PR . 12 Z 



2TT^ . dK G 

which for wrought iron, in which G 
PR L 



i3° = 0.00062 



a 
^36oS 12 L_ 

TT G d ' 

11,386,000 gives 
L 



^ 0.000120S S — 
a^ d 

For cast iron these values are doubled, giving 

ao PR L . ^ L 

tf° = 0.00124 = 0.0002436 o — 

d^ d 



(136) 



(137) 



• (137) 



Here L is taken in feet and ^Sis the stress at the point of ap- 
plication on the shaft. It will be noticed that the angle 1? can 
be determined very readily when .Sis known. It must be re- 
membered that d and 5 are closely related, and that the value 
of d depends upon the value taken for S. 

Various applications of twisting moments may be reduced to 
a single one for use in the formulas, by classification under some 
one of the following heads, taking the value for L as follow 
(see ? 13, ? 14) : 



94 



THE CONSTRUCTOR. 



(a). L = the whole length of the shaft, in feet, when the force 
is applied at one end and transmitted to the other. 

(b). L ^ half the length of the shaft when the twisting forces 
are applied over the entire length uniformly. 

(c). L = one-third the length of the shaft when the twisting 
forces diminish uuiformlj' from one end to the other 
of the shaft, as in § 14, case III. 

(d). In general, the distance of the point of application of a 
collected number of twisting forces distributed in 
any manner along the length of the shaft, may be 
found by multiplying the power applied at each 
point (in horse power), by its distance from the end 
of the shaft, adding the several products together 
and dividing by the total horse power transmitted. 

The methods may be illustrated by the same examples which 
were given in the preceding section. 

Example l. The screw propeller shaft given in the previous Example 1, 
gives the following: data : 



5= 8,200 lbs., d - 
According to (137) we have 



■ 15", i = 72 feet. 



i5 ^ 0.000121S 



8200 X 72 



15 



• = 4.75". 



This win be reduced to /g this value, or 3^° when either of the cranks is 
on the dead centre. 

Example 2. The lire of shafting given in H-^ample 3, of the preceding sec- 
tion, is made of two diameters in the bearings and in the body, and these 
must be combined. The bearings may be taken at 4 inches long each, and 
are 32 in number. We have tlien 



1^ =O.OOOI203 



[(■= 



■ o ■ 33 • 7397 \ , 
3.875 / ^ 



/ 223— 10)6541 \~l 



=: 0.0001208 (20158 + 348310) = 44 J^" ! ! 

a deflection which must be very marked, with variable loads, and entirely 
inadmissible with fine machinery. 
Example 3. If the preceding shaft had been made S inches diameter, as by 

N 
Fairbairn's formula, we have for =1.263. 



i^o. 



D. 00062 



62500 X 1:263 X 223 



2At. 



Example 4. In the twine factory at SchafFhausen there is a shaft made of 
Bessemer steel. The length is 4S9 feet, and it transmits 200 horse power 
from the Rhine up the bank and an angle of 23*^. The diameter is 4},]", 
N^ 200, n = I2D. This gives S= 4756, and if we take the luodulus of elasti- 
city of the steel the same as wrought iron, we have 

^o = „.^,,.8i^55^' = 58.34°. 

4.8125 

Example 5. A shaft 164 feet long, and of a constant diameter, transmits 70 
horse power at 100 revolutions. The power is taken off by a number of ma- 

N 
chines, ranged at uniform distances apart. According to the table for — = 

0.7 the diameter should be about 4^'i". In determining the torsion the value 
of Z. is taken at one-half the length of the line (case b) giving : 

62500X0.7X82 * £,,0 

i9° = 0.00062 ~ — ' ^= about 6^° 

Since the formula is based on an angular deflection of 0.75° per foot, we 
might have obtained direct, 82 X 0.75 = 6.35'^, or nearly the same value. 

If in any case the calculated deflection appears too great, the diameter of 
the shaft must be increased, and since the denominator of the equation is 
the fourth power of the shaft diameter, a slight increase in its value effects 
a marked reduction in the deflection. 

Example 6. If the angle in the preceding example is desired to be reduced 

to one-half its value, the diameter must be multiplied by v 2 or by 1.1S9, 
hence ^ = 4.25 X 1.189= 5 inches. 

?i48. 
Journals for Shafting. Round Roi.i,ed Shafting. 

The journals on shafting are either end journals, and treated 
as already shown, or, as in most cases, necked journals, and the 
length of bearing made as given in g 92. For line shafting the 
.special determination of journals is unnecessary. Unless there 
is some apparent reason for a special determination of the jour- 
nals (as in the case of locomotives), the journal length / is usu- 
ally taken quite large, as frf, 2d, \d (see § 109 ct seq.), care be- 
ing taken toinsurepropersupportof the journals in the hangers. 

The Kirkstall Forge Company, of Leeds, have prodticed shaft- 
ing which is rolled round and requires no turning. The round 
finish is given by the action of plane discs whose geometric axes 
are horizontal and parallel, about eight inches apart, andrevolve 
rapidly. (See 2 195.) The discs are placed so as to act upon 
the bar as it leaves the rolls, and are cooled with water, and 
their action produces a true cylindrical form to the shafting, 
and gives it a highly finished surface, so that it is at once ready 
for use without being turned in a lathe. By this process the 
modulus of resistance is also increased nearly 20 per cent, over 
that of shafting rolled in the ordinary manner, as shown by tests 
by Kirkaldy, and given in the catalogue of the Kirkstall Forge 
Company, for the Melbourne Fxposition. This feature is not of 
as much impoitance as at first appears, although it is of some 
value. 



The absence of turning is also of advantage, and the increas- 
ing use of this shafting is doubtless due to both causes. The 
principal objection to it lies in the fact that the hard outer skin 
cannot be disturbed without affecting the truth of its form. 
Keyways cut in it invariably cause springing. 

Some of the modern methods of sectiring pulleys without 
cutting keyways may be used to avoid this difficulty. The jour- 
nals and wheel seats on this kind of shafting do not require 
turning. 

? 149- 

Combined Sections. Wooden Shafting. 

The dimensions for shafting of various combined sections 
(tubular, cruciform, fluted) are determined by finding the size 
for round shafting of the same material, and then deducing the 
dimensions of the desired section in the same manner as given 
for axles in §§ 138 to 142. 

Axles of wood (generally oak) are made of polygonal section 
described about a circle not less than 1.75 times the diameter of 
a cast iron shaft for the same work, this being the fourth root 
of the ratio of the moduli of elasticity of the two materials. 
Wooden shafting is now seldom used. 

? 150. 
Shafting subjected to Deflection. 

Shafting is often loaded in such a manner as to be subjected 
to bending stresses, and as already seen, this is the most general 
condition in which it is used. Under these circumstances the 
combined resistance must be taken into consideration. This is 
most conveniently done by assuming an ideal bending moment 
(see I iS). 

Let M^ be the twisting moment for a given shaft section, 

M(j be the bending moment for the same section ; 

then the ideal bending moment combining them both will be : 

( AT, ),. = Vi M„ -f ^ V^TJ^rjQ .... (139) 

This formula may be simplified for numerical calculations by 
Poncelet's theorem, approximately : 

When M^ > iJ/^take {M^~)^ =0.975 M^, -f 0.25 M^ . . (140) 

and when 7J/^ > 71/^ take (.^4); =0.625 M^ + 0.6 M^ . . (141) 

An examination will be made, first by the analytical, and then 
by the graphical method. 

I. Analytical JMclhod. — The axle or shaft ABC, shown in 
Fig. 421, carries'a gear wheel R at C, which acts tangentially to 




■T"' 



Fig. 421. 



rotate the shaft with a moment Md ^ Q R, and also acts to bend 
the shaft with a force whose reactions are parallel to Q, and are 

Pi=— — at A, and P., = -^=-— at B. The greatest stress is 

a-\- s ' a-\- s 

at C, for there both bending moments are at their maximum 



Mb ^ 



P^_P, 



a 



, hence calculation should be made for this point. 



Example.— 'L&t Q= 5500 lbs. J\ = 11^^", a = iQ^^", J = 78^4'', then 

78 7^ 
= —^ 0= 0.8 = 4400 lbs. 
98.50 

I9-7S , 



/>„ = 



i.50 



i2 = 0.2 iQ = iioo lbs. 



Also 



^^d = 5SOO X 11-75 = 64,625- 
Mj^ = 4400 X 19.75 = S6,Qoo. 
M]^ ^^'^d and formula (140) is used. 
= 0-975 X 36,900 -1- 0.25 X 64.625 = 84,727 -I- 13,656 = 98,383" lbs. 



Hence 

We have (J>f/,)i 

From this the diameter at Ccan be calculated. If the shaft is of cast iron 
with cruciform sectiou, we have for the diameter D, 



and taking .S= 4250 -we have 






32 



oS3_X_32_ 

4250 TT 



-.6%" 



The journal at .-i is found in the table of §gi, column 4, to be about 2^". 
For the neck journal at B, we have from the table of g 145, taking the double 
value for cast iron, d-i = ^Vz"- 



THE CONSTRUCTOR. 



95 



Graphical Method. — The same example may be solved graphi- 
cally, lu Fig. 422, with a horizontal closing line, construct the 
link polygon a b c, for the bending moments, and the force 
polygon a 10, giving the forces P^ and />,, and also a c c' , the 
surface of moments for the shank A C. 

The moment ISId is yet to be determined. In the force poly- 
gon with a distance R from the pole O, draw a vertical ordinate ; 
this will be il/rf. Lay its value off at c'r,, and bl\, and 's of 
these values give c' c^ b^ b for the parallelogram of torsion for 
the shank CB. 




Pig. 422. 

The combination of the bending and twisting moments may 
then be made by formula (139). Make cc.^^ '/% c c' and join c. 
b, then at any point of the polj'gon, as for example at/, the 
distance// ^ Ys/f'- Now transfer c' c^ to a b, at c' c^' ; then 
will the hypotenuse of the triangle C2 c' c/ divided by c^ cj = 
■^{ficc')'' + (^iCtC^Y', and the sum cc, + c.^Ca'= c c, + ac^the 
desired moment {Mi,)t for the point C. In the same manner we 
obtain// -|-/2/„'=//,-|-//3 the moment {jVb)i for the point 
J^- The line ("3/ b^ is a curve (hyperbola) which may be taken 
approximately with sufficient accuracy as a straight line i\ b^,. 
The various dimensions may be obtained from the pol5-gou a c 
b b^ c^ c' in a similar manner as shown in the discussion of axles. 

Other discussions of this subject will be given when consid- 
ering rock shafts and crank axles. .. 

CHAPTER X. 

COUPLINGS. 

? 151- 

Various Kinds of Coupi<ings. 

The devices by means of which the different lengths of shaft- 
ing are connected together so that the motion may be trans- 
mitted from one piece to the next, are called couplings. 

They may be classed as follows : 

1. Rigid Couplings. 

2. Flexible Couplings. 

3. Releasing, or Clutch Couplings. 

The iirst class includes the various forms of coupling for line 
«hafting and the like, in which the coupling and the coupled 
portions have the same geometric axis. Flexible couplings are 
those which permit more or less change in the relative position 
of the coupled shafts ; while clutch couplings are constructed so 
as to be thrown in and out of engagement, usually when the 
parts are in motion. These three classes are all shown in vari- 
ous forms in the following examples : 

i, 152- 
/. Bigid Couplings. 

Rigid couplings may be made either in a single piece, or ia 
several parts. Of the first sort is the so-called Muff Couplings, 
Fig. 423. The muff is fitted over both pieces of shafting, and 
a single key binds the parts all firmly together. 

In giving the proportions of the various parts of the following 
couplings, we may take for a unit or modulus the thickness <J of 
the hub, making its value equal to : 



(5 = 



3 10 



(142) 



i/ being the diameter of the shaft, whether of wrought or cast 
iron. The dimensions of the key may be taken as given in J 6S, 
-Formula (71) for torsion keys. 



More recently, in exposed situations, the projecting end of the 
key is covered with a cap, in order to avoid accidents. 

The form of coupling shown in Figs. 1S9 and 190, I 69, looks 
very practical, but the test of prolonged use will be necessary to 
demonstrate its merits. 




Fig. 423. 

The simplest two-part coupling is the well-known plate coup- 
ling, Fig. 424, and its form permits the nuts and bolt-heads to 
be kept below the projecting flanges, and thus out of the way. 

The number of bolts in a plate coupling i = 0.8 rf + 2. The 
diameter d of the bolts should be 0.125 rf + -f/', which gives a 
strength pioportional to that of a shaft calculated by formula 
(133), or if d is determined from formula (133) the bolt will be 
strong enough.* 




Fig. 424. 

Plate couplings are extensively used in England and Germany, 
although they are being superseded by later forms. Their 
strength has caused them to be used for coupling the lengths 
of screw propeller shafts, and in this case the plates are forged 




Fig. 425. 

on the shafts, thus dispensing with the use of a key, Fig. 425. 
This form was introduced by Langdou in 1S52, and is in general 
use, 4 to 6 bolts being used. 

Examples : The following ca.ses will ser\'e to give the proportion of such 
plate couplings in executed designs. 
Jason. James Watt & Co., d = 12", D = 24", d\ = 3", b = 6", i = 4. 
Warrior, John Penn & Son, d— 17", D = 37'', di = 4'', b = 10, i = 6. 
Vessel by Ravenhill & Hodgson, a = 12", D ^^ 25, di =^ 3", b = 6'', i = 4. 

Fig. 426 shows a clamp coupling divided into two parts longi- 
tudinally. This form is provided with two keys, and the man- 
ner iu which it is bolted together. If it is desired to clamp the 



.2 jH,»+2.6d.; 




Fig. 426. 

shafts together endwise, the small circumferential grooves and 
lips may be used as shown. Such grooves may be used iu depth 
equal to o.oi d -|- xV. ^^^^t may be omitted where endlong clamp- 
ing is unnecessar}'. If lock nuts are used on the bolts the main 



* The dimensions in Fig. 424 are correct for English measurements, except 
the bolt diameter, which is as given above, and the distance from hub to in- 
side of flange, which should be 2.6 d + VI". 



THE CONSTRUCTOR. 



nuts may be counter-sunk as shown in the illustration. The 
rnimber of bolts is = 2, 4 or 6, rarely more, and of diameter as 
follows : 

i = 2, 4, 6 or more. 



d^=i+A 



d_ II' 
7 32 



S "^ 16 



Example ; For a shaft 2§'' diatneLer, with a coupling fitted with two bolts 
the dianieter d^ = 0.77", say -}{]", for fourbolts d^ = 0.72", say ;5", forsix bolts 
(fi — 0.67, say ly. 

This form of coupling has been made with bolts with differ- 
ential thread passing through both parts and giving increased 
clamping force.* 




Fig. 427. 

The cone coupling shown in Fig. 427 is the design of the 
author, and is a modification of the preceding form. The keys 
are cast in one with the halves of the inner cone, and are planed 
to fit the keyways in the shafts. The cone is made with a taper 
of jiy on a side, which will hold the parts securely when driven 
on, without any other fastening. If there is much vibration, 
however, it is advisable to have a screw thread cut on the inner 
cones as shown, and the outer shell tightened by a spanner. In 
most ordinary cases the screw may be omitted, and a small steel 
countersunk set screw tapped into each side of the shell to 
clamp the inner cone. If endless motion need not be considered 
the circumferential grooves may be omitted. 

With couplings for shafts larger than 2^-3'', the bearing sur- 
faces should be recessed to reduce the amount of finishing. 




Fig. 428. 

In America Sellers has introduced a clamp coupling in which 
two cones are opposed to each other and drawn together by 
three bolts, the whole being inclosed in a cylindrical shell bored 
out to fit the cones as shown in Fig. 42S. The cones are cut 
through on one side so that they are compressed by the action 




Fig. 429. 

of the bolts. A key is let into the cones and shafts diametri- 
cally opposite the cut in the cones. An especial advantage 
which results from the double cone construction lies in the fact 
that it is not necessary for the two shafts to be exactly the same 
diameter.t 



* Ruggles' Coupling, Pract. Mech.Joiir., 1866, p. 185. In Fig. 426 there are 
two dimensions which require transforming: for 20 -}- 2.6 rf, use ^" -^ 2.6 d, 
and for 10 -i- 1.3 d use %" -f- 1.3^. 

t Among other tests the Sellers Coupling stood the following; Two shafts 
10 leet long were fitted in three hangers, the middle hanger next the coup- 
ling being set i^^" out of liue, and after several weeks' time, at 250 revolutions 
per minute, the coupling remained intact. 



In England Butler's cone coupling has been used, and was 
designed for use with the cold rolled shafting described in g 14S. 
It is similar in construction to Sellers', the three bolts being re- 
placed b}' a single concentric screw thread and nut at each end. 
The key which Sellers uses is omitted in Butler's coupling, the 
shafts being held only by the clamping action of the cones. 

In the United States Cresson's coupling is also much used. 
Its construction is shown in Fig. 429. The clamping surfaces 
are cast in one with the outer shell, and forced upon the shafts 
by means of the tapering screws. This coupling possesses the 
same advantage as does Sellers', in being adapted to shafts of 
slightly unequal diameters. 

//. Flexible Couplings. 

§153- 
Various Kinds op Flexibi^e Coupwngs. 

Couplings which permit the shafts to change their positions 
may be required to meet three different conditions. The motion 
of the shafts may be : 

[a) I,engthwise, or in the direction of the axis. 

(5) Crosswise. 

(c) Angular, the shafts being inclined to each other. 

In some cases two, or even all three, of these conditions may 
be present. In the first place the axes of the two shafts coin- 
cide ; in the second case they are parallel to each other ; in the 
third case they intersect, while in the combination of {b) and {c) 
the axes pass each other. All these cases occur in actual prac- 
tice and meet with useful applications. 

?iS4. 

Couplings for Lfngthv^ise and Parallel Motions. 

Endlong motion of the coupled shafts may be provided for by 
giving various prismatic forms to the parts of the couplings. 
As an example, Sharp's Coupling, Fig. 430, may serve. This 
permits but slight movement lengthwise, and also a little angu- 
lar displacement, and is therefore suitable for positions where 




Fig. 430. 

the bearings cannot be accurately placed. In some recent ex- 
amples of this form of coupling, one half is first made and the 
second cast upon it, and in this case the outer recess is omitted. 
In some French screw steamships in which the screw is 
arranged to be lifted, one of the shaft couplings is arranged so 
that sufficient endlong motion may be obtained to permit the 
end of the shaft to be withdrawn from the hub of the screw. J 



30 OjSO^a 1,6 2,0 




Fig. 431. 

For shafts in which the displacement is crosswise, the axes of 
the shafts remaining parallel, Oldham's Coupling, Fig. 431, is 
most applicable. This consists of two end pieces and one inter- 
mediate piece. The latter has a prismatic tongue upon each 
side, the two being placed at 90° with each other, and fitting 
into corresponding grooves iu the end plates, which latter are 
keyed on to the shafts. If the two axes of the shafts coincide 



X Among various interesting examples of such couplings may be men- 
tioned those described in Armengaud's " Vignole des Mecanicie'ns," Paris, 
1863, Plate 9 ; also Ledieu, App. a\'apeur de navigation, Paris, 1S62, and Or- 
tolan, Mach. \ vapeur marines, Paris, 1859. 



THE CONSTRUCTOR. 



97 



at O, the tongues and grooves have no sliding action upon each 
other. If one of the axes is moved parallel to itself, say to P, 
the middle of the intermediate piece will describe a circle 
O Q P Q' oi the diameter O P= the distance between the axes, 
making two revolutions for every revolution of the shafts, the 
other points of the disc describe cardiode paths. The velocity 
ratio remains constant. 

Another form of coupling for the purpose consists of two 
cranks connected by a short drag-link to permit the necessary 
movement. This is frequently used in connecting engine shafts. 

? 155- 
Jointed Couplings. 

The best known of all the flexible couplings is the Universal 
Joint, known also as Hooke's Coupling, and also as Cardan's 
Coupling.^" This form of coupling permits the connection of 
inclined axes within certain limits. It consists broadly of two 
end pieces, and a middle piece, the latter containing two pairs 
of journals placed at right angles with each other in the form 
of a cross, each pair fitting into journals on one and the other 
of the end pieces respectively. 

The rate of transmission of motion is not uniform, and is 
dependent upon the angle of inclination a of the two shafts, 
the angular rotations u and u^ of the driving and driven shafts 
being expressed in the following ratio : 

tan (J, 

^ = cosa (143) 

tan 6) 

which gives a periodical variation whose period is lSo°. The 
following table gives the values of «j for successive values of u, 
for various angles a : 




For small values of a the variation is unimportant. For the 
angular velocities a, and Uj, we have the relation ; 



which gives a maximum 



I — sm"^ u sva.'- a 
I 



(144) 



• and a minimum cos a. These 



variations in velocity maj' be neglected when the moving masses 
are inconsiderable and the angle a is small f 




Fig. 4 3 2. 

The detailed construction of the coupling admits of great 
variations. Fig. 432 shows a form with cast-iron end pieces and 

wrought iron middle piece. The relation — is varied also. The 

journal diameter d,^ is determined by the methods already given, 
and from the moment of rotation {P P) the journal pressure P^ 

may be taken with sufficient accuracy as ^ -^ — = — . The dis- 

tance a should be made greater as the angle a is increased, a 
being made quite in the illustration. The joints in the boxes 



* If not the original inventor of the Universal Joint, the Italian, Cardan, 
was the first to describe it {i5oi-i576),andthe Englishman, Hooke (1635-1702), 
first applied it for the transmission of rotary motion. 

f In the table the values of cu are so placed that when oj ^ O, the cross 
journals of the coupling lie in the plane of the shafts. 



should be made in the plane of the shafts, not at right angles 
to them, in order to provide for the wear. 

Universal joints are used to good advantage in screw propeller 
shafts in order to provide for the flexure due to the elasticity of 
the hull of the vessel. In such cases two universal joints are 
used on a shaft. A coupling for such service is shown in Fig. 
433. Here all these pieces are forgings, one end piece being 
forged solid with the shaft. The middle piece is formed of a 
double ring, the bearings being held between the two parts. 





Fig. 433- 

while the journals are secured to the end pieces. No provision 
is made for taking up the wear upon the journal boxes, as the 
angle a is so small that the wear is very slight. The length 4. 
of the journals need not be great, about i to 1.25 d^, and since 
J*? can be kept small, the dimensions of the entire coupling may 
be kept withui reasonable limits. 

Another form of universal joint is shown in Fig. 434. Here 
the cross journals are made in the form of through bolts, pass- 
ing through both end and middle pieces. This requires a slight 
modification in the form, since the axes of the two bolts cannot 
intersect each other. A slight error in motion follows, causing 




Fig. 434. 

a very small endlong motion with each revolution in the coupled 
shafts, but this is generally insignificant. This form is suitable 
for agricultural machinery, horse powers, etc. In the illustra- 
tion a proportional scale is given. Tlie modulus is as before, 
S=z\d+ iV'. • The pole P is therefore taken, so that ^ d -\- 
JL" = o, or so that fl' = — ^r," ■ 

The irregularity in motion shown in formula (144) is generally 
of little consequence, and need not be considered except iii 
cases requiring geometrical exactness, as in the connections for 
large tower clocks, or in cases where large masses are drawn at 
high velocities, as in threshing machines. The variation can 
be obviated by the use of a double universal joint, which con- 
sists of two simple couplings. If, Fig. 435 a, the driving shaft 
A is coupled to B by means of a short intermediate shaft C^ 
the connections being made by two similar couplings of the 
same angles, then the motion of A will be transmitted to ff 
without variation. In this case the driven shaft may be given 
several positions with regard to A, being placed at B, making- 
the angle 2 a, or at B', parallel to A, or at B" on the surface 
of a cone of half the angle a, which is made with the inter- 
mediate axis C The two universal joints are similarly placed 
when the cross axes belonging respectively to A and B lie in- 
the same plane as the shaft C at the same tunc. In the posi- 
tions B and B' all three shafts He in the same plane, but not in 
position B". In the last case A and B intersect. 



98 



THE CONSTRUCTOR. 





Fig. 435. 

If .the cross axes are not placed similarly, but, for example, 
at 90°, as in Fig. 435 b, the variations of motion are increased, 
.and we have. 

tan Uj = tan o cos'^ a, in which tj and "i stand for A and B. 

"If a = 30°, we have for w = 45°, tan w, = (i "^ 3 )'' = o-75, hence 
'6)j = 36° 54' instead of 40° 54', as in the table above. 

A concealed form of universal joint is that used in rolling 
mills, the cross being formed upon the end of the roller. 




Fig. 436. 

Fig. 436 shows a form of link coupling designed by the 
author. A is the driving, and B the driven shaft, the arm C 
being jointed at 3 by journals at right angles to B, while at 2 it 
slides in a bearing rigidly attached to A. The axis B makes 
with A an angle a, and the piece 3 C2 makes with A the 
angle 90° — pj. For the relation between the angular rotation 
<jj and (J^ of the shafts A and B, which are supposed to revolve 
in fixed bearings, we have : 

sin a tan /? \ ] 



tan u, 



: tan 



"ii 



tan U4 = - 



cos a cos <Jj — sin a tan /3 



(145) 



In this case the transmission of motion is much more irregtdar 
than with the universal joint, since the angular velocity ratio 
between 



cos a -I- sin a tan /3 



and 



I + sin- a tan /i 



■fluctuates. This coupling is really only a modification of the 
universal joint, inclined to such an extent that the fork of the 
shaft A stands at right angles to the axis. (Compare formula 
(144) with (145)). By combining two such couplings symmetri- 
cally with each other, as in Fig. 437, the motion will be uni- 
formly transmitted. The two sleeves at 3 and 4 are formed in 





Fig. 437. 



Fig. 438- 



one piece, their axes making the angle 2 /3 with each other. In 
practice it is better to make these parts in the form of journals 
and place the sleeves in the C, and C, Fig. 43S. These parts 
are also prolonged beyond the shafts in order to counterbalance 
the weight. The pieces 3 and 4 can be bolted firmly together, 
since their relative position to each other is constant. It must 
"be observed that a must not exceed 90 — /3, otherwise a dead 
point will occur. 

The parts 3 and 4 may also be connected by ball joints 3' 4', 
Fig. 439, in which case the device becomes the coupling of 
■Clemens.* Here the counter-weights are omitted, and the parts 



Cj and C, connected on both sides. Clemens has used this form 
with the angle 2 a = 90°. The doubling of the parts has the 




Fig. 439- 

objection that it requires more accurate fitting of the parts than 
where only one side is connected. If the axis B is placed par- 
allel to A, as at B' in Fig. 437, the rate of motion will be very 
irregular, for a= /3 = 30° as in the illustration, the velocity 
ratio will vary between yi and 2. 




Fig. 440. 

In many screw vessels a simple form of flexible coupling is 
used, suited for slight angular variations. In Fig. 440 this is 
shown, and it will be seen to give slight flexibility similar to the 
universal joint, and sufficient for many cases. A bearing should, 
be placed back of the coupling on each shaft. 



///. Clutch Couplings. 

I 156. , 

Toothed Clutch Coupungs. 

Couplings of this form may be distinguished by their method 
of engagement, the clutch surfaces entering in and out of en- 
gagement axially, radially or inclined. 




Fig. 441. 

The oldest form of clutch coupling, and one of the most 
widely used, is that shown in Fig. 441. Here the engagement is 
axial. The modulus for the proportions is the same as before, 
'5 = -J- (f -j- j\" ; and an approximation to the number of teeth 
may be given by making z^ i -\- 0.6 d. The clutch is thrown 
in and out of gear by a lever which works in the groove in the 
portion of the clutch on B. Examples of suitable lever forks 
are shown in Fig. 442. 




* Clemens' Angular Shaft Coupling, U. S. Patent, Nov. 10, 18 



Fig. 443 

Various forms of clutch teeth are used. The forms in general 
use are given in Fig. 443. The first form is adapted for motion 
in either direction, but can only be operated when moving 
slowly. The second form is more readily thrown into action, 



THE CONSTRUCTOR. 



99 



tjut is adapted to transmit motion only in one direction. The 
■driving faces are inclined very slightly, from the normal to the 
direction of motion, the angle not being enough to cause any 
tendency to disengagement. In the third form the teeth are 

1 I • 1 





Fig. 442. 

-more blunt in shape at the point, which adds to their strength 
against breakage when subjected to shock. The last form is a 
combination of the preceding varieties, and like the iirst, may 
be driven backward. In spinning machinery, light couplings 
with many fine teeth are used and operated at high speeds. In 
some screw vessels in which there is no provision for raising the 




Fig. 444. 

screw, it is desirable to disconnect it when proceeding under 
sail alone, and some form of clutch coupling is used. A very 
simple form is the so-called " cheese coupling," used in English 
vessels. Fig. 444. The hub of the propeller is provided with a 
bearing on each side, and formed with a T projection fitting into 
a corresponding recess in the heavy flange (or cheese) on 
the shaft A. The propeller blades are secured to the hub as 
already shown (Fig. 191). 

I 157- 

Friction Clutches. 

Couplings in which one portion transmits motion to the other 
portion by means of friction, are often especially applicable, 
since by the mere removal of the frictioual contact the parts are 
disconnected, and when they are thrown into contact the driven 




Fig. 445- 

portion is put into motion gradually. By making friction coup- 
lings of large diameter, they may be used to transmit propor- 
tionally great rotative moments. In Fig. 445 is shown a friction 
coupling used by Ramsbottom in rolling mill work.* The part 
A is firmly clamped between the wood-lined surfaces of B ; but 
the parts may be arranged so as to slip if undue resistance is 
encountered, thus making it a safety coupling. The modulus 



as before is S 



- + 



1 6' 



Cone couplings are used also, in many forms. In the example 
shown in Fig. 446 the driven portion A of the coupling carries 
a gear wheel shown in the dotted lines, to which motion is to 
be transmitted from the shaft. The two parts are forced into 
engagement by the screw and hand wheel b. If the parts are 
so arranged that the motion of the hand wheel b is in the same 
direction as the rotation of the part B, when the latter is thrown 
into engagement, it is only necessary to hold the wheel b sta- 
tionary in order to throw the clutch out of gear. From the mean 
radius R of the cone surface, and the angle of taper n, we have 
for an axial pressure Q, for any circumferential force P: 



nf sinn , 
= P\ —^ + cos 



■y 



(.46) 



in which y is the coefficient of friction between the cone surfaces, 
and (PR) is the statical moment tending to rotate the shaft. 



I f sin a , \ 

/ + cosn I 




Fig. 446. 

The angle n should not be taken at less than 10°, in order '.^at 
the parts may not become wedged together; for iron on I'ron.y" 
may be taken at 0.15. In order to keep both Pa.ud Q as small 
as possible, P should be made large, say between 3 and bd. 

The relative motion of the screw and hand wheel is of course 
dependent upon the radius of the wheel b, and upon the pitch s 
of the thread. 

Example ; A wrought iron shaft of a diameter f^= 2 inches, making 50 rev- 
olutions per minute, would transmit, according to the table of 'i 145.0.0313 X 
50 = I 5 H. P., or have a statical moment P R = 1975 inch pounds. If the ra- 
dius of the friction cone couple is equal to 5^, or 10 inches, we have, accord- 
ing to (147) Q=~^ / — ^--(-cosaj. If a = 10°, and/™ 0.15 we have 

£•=197^ 5 ( °'J"^ 4- °-984S ] = 423 lb?. 

Suppose the hand wheel to have a radius of 4 inches and the screw a pitch 
of /^"j we then have for the force to be exerted on the rim of the hand wheel : 



0.25 X4g3 .. 
"X4 



.8.4 lbs. 



For the transmission of moderate force the cone coupling, or 
some of its various modifications, has very generally been used.* 




Fig. 447- 

Instead of a single pair of external and internal cones, a num- 
ber of small elements may be employed. This form is shown 
in Fig. 447. The general calculations are made as above, except 
that the lever arm R of the friction must be reduced, and may 
be taken with sufBcient accuracy at a point distant from the 
outer circumference equal to one-third the width of the grooved 
frictioual surface. The operating lever in this case need make 
but very little movement, and the arrangement of a fork mounted 
on an eccentric bearing, as shown in the illustration, may be 
conveniently adopted. t 

When a cone coupling is intended to be used for the trans- 
mission of large forces, the apparatus for pressing the parts to- 
gether may sometimes be so arranged that it is mounted on 



*See Kngineer, 1866, January, p. 44 ; also Genie Industrielle, Vol. 32, p. loi ; 
and an older form of this coupling is shown in Salzenberg, Vortr. p. 173. 



* Many such applications will be found in the description of the Suez Ca- 
nal ; see Armengaud. Publ. Ind., Vol. 17, PI. 9. 
fSee Armengaud, Vignole des Mecaniciens, Plate II. 



lOO 



THE CONSTRUCTOR. 



the shaft, revolving with it, without creating so much pressure 
against the bearing. The fork and grooved collar shown in Fig. 
447 is uot suitable for heavy clutches on account of the excessive 
collar friction, hence the pressure is better applied by means of 
a screw mounted on one of the shafts, and this may be conve- 
nientl}' arranged so as to draw both shafts firmly together. Sup- 
pose the shaft to be 4 inches in diameter, we have from the pre- 

24 V 015 



ceding, i?^ 6^/- 24", and an axial pressure Q- 



0.984S '\ = 



2S1S lbs. This endlong pressure, instead of creating 



hurtful collar resistance, may be utilized by arranging the parts 
as shown in Fig. 44S, which shows a friction clutch coupling of 




Fig. 44S. 

the author's design. As shown in the section, the part A ex- 
tends over the part B, and both parts are drawn together by the 
action of the screw and hand wheel. The only modification in 
the screw gear is that the screw is made large enough to permit 
the shaft to be passed through it, the thread being thus cut upon 
the hub of the part A. This coupling runs very smoothly. The 
concentric channels should be arranged with clearance at the 
bottoms of the grooves, as shown in the section, to provide for 
fitting and wear. The modulus for the parts is the same as be- 

3 i6- 

In Fig. 449 is shown the cylinder friction clutch of Koechlin. 
In this case the clutch movement takes place radially. The part 
^ is a hollow cylinder in which three internal clamp pieces are 

f> -"I _f"] • 



fore, viz., <5 = 




Fig. 449. 

fitted, each being provided with a bronze shoe. These are thrown 
in and out of action by means of a sliding collar £', which 
operates right and left hand screws by means of the lever d. 
The clamps slide in radial grooves and the details are fully shown 
in the illustration. The nuts for the right and left hand screws 
can be closely adjusted and clamped by set screws, so that a 
radial movement of less than y'^" 1== sufficient. There is no 
danger of wedging the parts fast in this form of clutch, as may 
be the case in cone clutches, as the elastic reaction of the cylin- 
der assists in the direction to release the parts. At the same 
tinJe the screws prevent the coupling from releasing itself and 



2S 



the axial pressure Q, upon the collar B', can be transmitted so 
that the screws need not have too quick a pitch. 

If s is the pitch, b the length of lever arm,ythe coefiScient of 
friction of the clamping pieces, we have for the transmission of 
a given moment {P -R), neglecting the friction of the screws, 

P ^ s PJ? , . 

7°'^=-i JR ('47> 

which gives a very small value for 0. 

If the parts are so arranged that B is the driven part, there 
will be no collar friction at B' , when the coupling is not in ac- 
tion. When the shaft is vertical, a weight may be used instead 
of a collar and lever, and by gradually lowering it the apparatus 
may be started with very little shock. The first clutch made by 
Koechlin was designed for the transmission of 30 H. ?."■' The 
above value corresponds to a minimum value of R. The modu- 



lus is the same as before 



3 ^16 



A verj' excellent form of this coupling was designed by Bod- 
mer, independently of Koechlin, f and a similar arrangemen.* 
has been adapted to mill gearing with success, j 




Fig. 450. 

Cylinder couplings in which the clamping pieces are operated, 
by toggle joints are also made. An example is shown in Fig. 
450, which is a clutch by Fossey, as applied to mint machinery .§ 
This is a very compact design and is arranged with four clamps, 
which have no bronze shoes. The toggle links are as wide as 
the clamps and are fitted with half-journals to transmit the pres- 
sure outwards, while to draw the clamps back, light through 
bolts are used (see ? 95). If the toggle links make with the axis 
an angle 90° -|- o, we have for the axial collar-pressure : 

^ Ptan a. PR tan a , „, 

0= = (148) 

^ f R f ^ ' 

The angle a may be taken very small, since there is no danger 
of clamping. The value may be as small as a = 2°, or even 1°. 




Fig. 451 



For a = i^^° we have 



Q _ 0.03 I 

~P " "a 15" ~~ T 
Another form of cylinder coupling using toggle levers, has^ 



* Bulletin von Mulhausen, 1S54, p. 138. 

t See Fairbairn, Mills and Millwork, Vol. II., p. 92. 

X See Uhland's Prakt. Masch Konstrukteur, 1S69, p. 97. 

I See Armengaud's Publ. Industrielle., Vol. XVII, PI. 10. 



THE CONSTRUCTOR. 



lOI 



been designed by Garand.* Jackson uses hydraulic pressure to 
force the clamps into contact.f Dohmeu-Leblauc uses springs to 
throw the toggles out of actiou.j Schurmann uses, instead of 
the separate clamps, a ring, which is compressed externally ;g 
Napier also uses a ring, expanded from within. || Becker ar- 
ranges the clamp blocks to be operated by centrifugal force.^ 
These are only a few of various modifications of the cylinder 
■coupling. 

A form of axial friction coupling which acts with very slight 
pressure is the Weston clutch, made by Tangye.** This acts 
upon the principle of multiple plate friction jsee \ loi), as is 
shown in Fig. 4,51. 

The wooden discs are engaged with the case, and the iron ones 
■with the shaft. In the form shown the plates are pressed to- 




Fig. 452. 

gether by the springs, and released by drawing back the collar 
B and releasing the spring pressure. A larger example of 
Weston's clutch is shown in Fig. 452. ^ is a winding drum, B 
the shaft driven by the engine. The outer disc C, and the inner 
•discs of the coupling are held apart by spiral sprinos, as shown 
at a. A light pull on the cord c holds the drum stationary ; a 
strong pull engages the clutch for winding; if the cord is left 
slack the load on the drum runs backward. 

I 158. 

AuTOM.\Tic Couplings. 

When power is transmitted to a shaft from two different 
sources, as from two independent engines, it is desirable to have 
one or both of them connected by a coupling which will auto- 
matically release or engage with the shaft, according to the dis- 
tribution of work. If one motor tends to overrun it will then 
be given more of the work, and so the resistance will be equal- 
ized. Such a device is the coupling of Pcuyer-Ouertier, gener- 
ally known as Pouyer's Coupling. 




Fig. 453- 



This is shown in Fig, 453. In this case the parts are so dis- 
posed that the part A, which is driven by one source of motive 
power, is loose on the shaft B, This part A maj' have gear 
teeth upon its circumference, for example, or may have a gear 
wheel mounted upon its hub, as shown by the dotted lines ; the 
hub being bushed with bronze. Upon the shaft B a ratchet 
•wheel is keyed ; the pawls a, a, being upon A, engage with the 



* Dingler's Polyt. Journal, Vol. 149, p. 22. 

fDin^ler's Polyt. Journal, Vol. 153, p. 251. 

t German Patent, 16Q52. 

?Zeitschr. d. Vereins d. Ine;., Vol. V, 1861, p. 301. 

ll Engineer, 1868, July, p. 64. 

^German Patent, 720:;. 

*»ln the U. S. by the Yale &. Tovvne Mfg. Co.! 



teeth when A drives B, but if B gains upon ^, or ^ stops while 
B continues to move, the pawls are thrown out of action. The 
direction of motion is shown by the arrow. The pawls are re- 
leased by the action of th- friction bauds b^ and A^, which are 
carried forward by friction upon B, whenever B gains upon A, 
the levers b throwing the pawls a out of gear. As soon as the 
limit of travel of the levers b, b, is reached, the friction bauds 
h^, b., slip upon B, being able to move no faster than A. When 
the speed of A increases and gains upon B, the pawls are again 
thrown into gear and A is automatically coupled to the shaft. 
In order that the pawls may not bind upon the ratchet teeth in 
releasing it is necessary that the angle 7, v/hich the pawl makes 
with the face of the ratchet tooth, must be less than the com- 
plemeut of the angle of friction ; iu this case y = 6o°. Pouyer 
uses only one friction band and makes both pawls engage at the 
same time. In the illustration the ratchetwheelis made with an 
odd number of teeth (13), and the pawls are placed so that a 
movement of only )4 the pitch will cause the parts to become 
engaged. The above proportion of the angle of the teeth is 
of importance, as otherwise the points of the teeth are apt cobe 
broken. The pawls also should be of hardened steel. 




Fig. 454. 

In Germany Uhlhom's Coupling is used for similar service, as 
shown in Fig. 454. Here A is the part connected to the motor, 
and B is fast to the driven shaft. A is an iuternal ratchet wheel 
into which the pawls 6 enter. The springs a serve to insure the 
entrance of the pawls into the teeth, which engagement continues 
so long as a drives B. If the speed of A is retarded, the pawls 
are retracted as shown in the lower part of the figure. In this 
case the springs act to keep them out of gear, being the reverse 
action to that of an ordinary ratchet gear. 

The pawls are fitted with half-journals (see ? 95), and are held 
in place bj' a plate ring, as shown. Ulilhorn originally used 
only two ratchet teeth in A, but increased the number afterwards 
to four, so that the parts would engage in a movement of one- 
fourth a revolution. It is better to use an odd number, as three, 
and by proper spacing of the pawls the greatest play will be one- 
half a space, or one-sixth a revolution with three teeth, as in 
the case of Pouyer's Couplings. B may be the driving part in- 
stead of the driven, but in that case the direction of the arrow 
must be reversed. 



CHAPTER XI. 

SIMPLE LEVERS. 
2159- 

jouRN.\i,s FOR Levers. 

In machine design a simple lever, or rocker arm, is a lever 
arm which is mounted upon an axle or shaft, at the end, about 
which it moves, and carries a journal upon the other end. For 
the proportion of the journal see Chapter V. The forms which 




Fig. 455- 

may be given to such journals are shown in Fig. 455, and are 
single overhung, double, or forked. The manner of securing 
the pin in the hub or the lever is most important. The pin 
should not be driven in up to the shoulder on the taper, but 
sufficient space left to insure that the fit is tight in the taper. 
This clearance is shown plainlj' in the figure. The same result 



I02 



THE CONSTRUCTOR. 



may. be attained by counter-sinking the collar into the hub on 
the lever. In the case of double overhanging pins, care should 
be taken that the load is equally divided between the two sides, 
so that the pressure upon each pin shall be equal to )A P. In 
the fork-ended lever the fit on both ends of the pin should be 
portions of the same cone. 

Example i. Vor P = 4400 lbs., we have from the table in ggo for alternating 
pressure and wrought iron journal, the diameter d= 0.027 v^44oo = i-S", and 
the length the same. For steel, we have 1^ = 0.024 -y/ 4400 = 1.6", and the 
length /= 1.3 X 1.6 = 2. oS". For a forked lever, a wrought iron pin with the 
same load the diameter, according to (9S) would be rf= 0.0185 s/ 44oo = 1.2", 
and the length / = 3.5 X 1.2" = 4.2"- 

All levers are not subjected to alternating pressure, but have 
the pressure constantly iu one direction, as for e.'^ample, the 
beams of single-acting pumping engines, etc. In such cases 
larger journals are needed. 

Example ■2. A wrought iron journal for a forked lever, under constant 
pressure of 4400 lbs., according to formula (98J, should have a diameter £/ = 
0.0212 \/ 4400= 1.4", and length 1= 3rfj= 4.2''. If the material had been cast 
iron we should have h_ad^=o.29 \/440o = 1.92", say 2", and 1=6". For steel 
■we have d = 0.0 185 \/ 4400 = 1.2", l^ j,d^ 4.8". 



2.J_ 




Fig. 456. _ 

2 160. 

Cast Iron Rock Arms. 

Rock arms may be either of cast or wrought iron. The hubs 
for wrought iron arms are given in the preceding illustrations, 
and in Fig. 456 are given some proportions for the various parts 
of cast iron arms. A fork-ended arm is shown below, among 
the walking beams, or if the fork hub is on the main axle, see 
the rules already given under Axles, Chapter VIII. 

§i6r. ■ 
Rock Arm Shafts. 

The axle upon which a rock arm works is usually subject both 
to bending and torsional stresses. The methods of calculation 
for all important cases are given in Chapters VIII and IX. The 
case which occurs most commonly is the overhung rock arm at 
the end of a shaft, and this is here given a special examination. 

If we have a, the distance between two planes normal to the 
axis, and passing through the middle of the pin and the middle 
of the bearing on the shaft, Fig. 457, there is an ideal bending 
moment with a lever arm J?, acting upon the bearing of the 
shaft, for a load P on the pin equal to 



{Mb 



i=:.Pa' = p( 



8 



-x/A'^ + <zM .... {150) 
See I 150. 
The lever arm a' is readily obtained graphically, as is shown 



in the illustration. For its numerical determination we have, 

a' = 0.62s a + 0.6 P L (151) 

and ii R<^a C 

a' = 0.957 a + 0.25 R J 
The lever hub must be made strong enough when the shaft is 
only subject to torsion, or when it is also subject to bending. 




Fig. 457. 

For wrought iron shafts wrought iron levers should be used, and 
for cast iron shafts cast iron levers. 

Let: 

w = thickness of metal of hub, 
7u = length of hub, 
P) = the shaft diameter for the statical luoment PP of a 
lever of the same resistance, see (133) and (134). 



for 



w 
T 

w 

15 



I 

25" 



(152) 



: 0.45 0.42 0.40 



If a lever is to be fitted to a shaft of greater diameter than D, 
we first determine the imaginary value of D, and insert it in 
(152). The same method is adopted if a cast iron lever is to be 
used with a wrought iron shaft, and vice versa. The shape for 
cast iron levers is given above, in Fig. 456. 

Example 1. If the lever of Example 1, ? 159 is made of wrought iron, and 
is 24 inches long, its statical moment /" J? = 24 X 4400 = 105600 inch pounds. 

This gives, from (131) D =0,091 -^105600 = 4.3", and if we take — - = ^^, we- 

have from (152), 2£/ = o.45 X4.3"= i-93". ^= 1-93" X 2= 3-86", say 3%". 

The hub may also be calculated of such dimensions as to bv 
strong enough to be forced on cold, and thus obtain sufficient 
friction to hold without the use of a kej' (see \ 65, formula 66)^ 

The friction O of the hub upon the shaft must then be ^( 






in which D' is the diameter of the shaft at the point where th& 
hub is fitted. 



D^ ^ D and 



49ti6 



Example 2. — In the^case] 01 the same levcrjjas the preceding^^example 
PR 105600 

We may then take Q = 50,000 and let / = A = 3^" and S» = 10,650, and sub- 
stituting in formula (66), we get : 



D 



>/ T X 4-3 > 



X 3.875 X 0.2 X 10,650 4- 50,000 



X 4-3 X 3-875 X 0.2 X 10,650 — 50,000 



^ 1/ f I6I500 - , , 

The key is used as an extra precaution for security. 




Fig. 458. 

A special method of keying, especially adapted for the hubs 
of levers and wheels, has been designed by engineer Peters. It 
consists of two parallel systems of keys, as shown iu Fig. 458. 



THE CONSTRUCTOR. 



103 



The taper of the keys is t,',,. The arrangement shown at (a) is 
preferable, as it weakens the hub less than {b). The angle a 
may be taken = 135°, the thickness of ke3's b = y'j D', and mean 
width h= 2.b. 

The form {a) is especially suited for hubs which are made in 
two parts. 

Those hubs which are upon shafts subjected to bending, are 
cousidered under the heading of Combined Levers, in Chapter 
XIII. 

I 162. 
Lever Arms op Rect.\ngui<ar Section. 

The calculations of the dimensions of simple lever arms of 
rectangular section are made upon the assumption that the force 




Pads in a plane, passing through the middle of the arm. Fig. 
459, and in a direction normal to the arm. 
If we let 

h = width of the arm at the axis, 
b = thickness of the arm at the axis, 
S^= the maximum permissible stress, 



= 4250, 







b = 


= 6^^ 

5/;^ 




Taking S for wrought iron = 


= S500, ar 


d for cast iron 


we hav 


e 
for wrought iron. 

^^ 

6 = 0.00072 — —- 

li- 






for cast iron. 

PR 

0.00144 . 

h- 



(153) 



These formulse are adapted for the determination of b, when 
h has been selected, the latter being most conveniently chosen 
with regard to the other condition. 



Example i. — Let ^=4400 lbs., 
'Jsl=-jy^' we have from (153) ; 



R=2^" for a lever arm of wrought iron, aud 



4400X24 



{7-I25)- 



If b is kept constant for the whole length of the arm, the 
width at the small end may be 0.5/;, while if a constant ratio of 
b : /; is kept, the small end = -3'/; (see ? 10, Case III and VII). 

If the force P does not act in the middle plane, as often oc- 
curs, then there must exist a combined bending and twisting 
stress on the arm. We may then derive a combined stress whose 
bending moment will give an ideal arm R'. 

If the plane in which the force Pacts is distant from the mid- 
dle of the arm by an amount c, we may make approximately, 
(see ? 150) : 



(•55) 



R'=i/^R-\-y^^R-^+c-' 

R'=o.s7S y?+ 0.25 c 

R>c, 

7?' = 0.625 R+0.6C 

Rye. 

may be determined readily by the graphical method. Fig. 
The third case shows the method for inclined arms. 



or 
if 

and 
is 

R' 
460. 

Example 2. — In the case of the lever of the preceding example, let C= 
I5-75''- This gives R ^ Cand we have from {154) ; 

R' = 0.975 X 24 + 0.25 X 15- 75 = 

= 23-4 + 3-94 = 27.34'' 

This gives for 6, 

4400X27.34 



b = 0.00072 



1-7 



(7-125)- 

Cast iron arms are sometimes made of cruciform section, see 
Fig. 456, in which case the ribs may be neglected. 



« 



163. 



Lever Arms of Combined Section. 

The sections shown in Fig. 461 are designed to secure an 
economy of material. Their dimensions are readilj- determined 
by first calculating a corresponding arm of rectangular section, 
and then transforming it into an I section, or double II shape. 
If //q be the depth aud b^ the breadth of the equivalent rectan- 



gular arm, and li and b the corresponding terrcs to be fouud, as 
in Fig. 461, we have 

A — _'_ 
ba~ i-\-a 
in which I . . (155) 



(I-OCx-Kt)] 



B 



These formulas permit a choice of the ratios -7- and -7-, which 
may be left to the judgment of the designer. In (155) the angle 




Fig. 460. 

irons of the third example in Fig. 46; have been neglected, and 
may be considered as making up for the weakening of the rivet 
holes. The following table .gives a series of values for (155) 
which will simplifj' the calculations materially. The table will 
also be found useful for other purposes, as all sorts of beams^ 
crane booms, etc. 




-c ■ '^ -^ ■■ - 



Fig. 461. 



i 161. 



Table for Transforming Arm Sections. 



// 








Values of 


I 

I J- 


2 








c 


B 
0.50 


3 
0.43 


3-5 
0.38 


4 


4-5 


5 


6 


7 


8 


lO 


6 


0.33 


0.30 


0.27 


0.23 


0.20 


0.18 


0.14 


7 


0.52 


0.45 


0.40 


0.35 


0.32 


0.29 


0.25 


0.21 


0.19 


0.15 


8 


0.54 


0.47 


0.42 


0-37 


°-34 


o3t 


0.26 


0.23 


0.20 


0.16 


9 


0.56 


049 


0.44 


0-39 


0.36 


0.33 


0.2S 


0.24 


0.22 


0,18 


10 


0.58 


o.,Si 


0.46 


0.41 


0-37 


0.34 


0.29 


0.26 


0.23 


0.19- 


II 


0.60 


O..S3 


0.48 


0.43 


0-39 


0.36 


0-31 


0.27 


0.24 


0.20 


12 


0.62 


O..S.S 


0.50 


0.44 


0.41 


0.37 


0.32 


0.29 


0.26 


o.2r 


14 


0.64 


0.5S 


0.,S2 


0.47 


0.44 


0.40 


0.35 


0.21 


0.2S 


0.23 


16 


0.67 


0-60 


o.s'; 


0.50 


0.47 


0-43 


0.38 


0-34 


0.30 


0.25 


lH 


0.69 


0.63 


O..S7 


0.52 


0.49 


0.46 


0.40 


0.36 


0.33 


0.27 


20 


0.71 


0.65 


0.60 


0-.S5 


0.52 


0.4S 


0.42 


0.3S 


0-,34 


0.29. 


22 


0.73 


0.67 


0.62 


°-S7 


o.,S3 


0.50 


0.45 


0.40 


0.37 


0.31 


24 


0.75 


0.68 


0.64 


0.59 


0.56 


0.52 


0.47 


0.42 


0.3S 


0-3,3 


27 


0.76 


0.71 


0.66 


0.62 


o.,=i8 


0-5.') 


0.50 


04s 


0.41 


0-3.5 


,30 


0.7S 


0.73 


0.68 


0.64 


0.61 


0.57 


0.52 


0.47 


0.43 


0-37 


ss 


0.79 


0.7s 


0.70 


0.66 


0.63 


0.60 


0.54 


0.50 


0-45 


0-39' 


^6 


0.81 


0.76 


0.72 


0.68 


0.65 


0.61 


0.56 


0.52 


0.48 


0.41 


40 


0.83 


0.78 


0.74 


0.70 


0.67 


0.64 


o.,58 


0-54 


0.50 


0.44 


4S 


0.S4 


0.80 


0.76 


0.72 


0.69 


0.66 


0.61 


0-.S7 


0-.S3 


0.47 


5° 


0.85 


o.Si 


0.78 


0.74 


0.71 


0.68 


0.63 


0-59 


0.56 


0.49 



Example 1. A lever arm has a length i? = 78.75" and the journal pres- 
sure at the end = P = 5500 pounds. It is to be of cast iron of double Tsec- 
tion with a height /iq = i2^s' '■ According to (153) we have for a rectangu- 
lar section 



Bq ~ 0.00144 



5500 X 78-75 



-= 3-9 



(12.625)2 

This is also so thick as to be impracticable, and hence the double /'sec- 
tion may be compared. Here we may take c \ n =^ \ : \z.^ B \ b = ^, and we 

get from the table =^ 0.44 and h — 0.44 b^ =- i. 71", aud the flange 

r + J 



104 



THE CONSTRUCTOR. 



breadth ^ = o. 44^ = 1.71 X 0.44 = 0.732, the web thickness = c = ^ = /: = 
— '- — ^ " 1.05", all of which are practical dimensions. It maybe found de- 
sirable to have c= b, or any reasonable ratio for B : b, and ^ ; /* be chosen. 
Example 2. A wroug^ht iron arm lias been found to require b^ = 2%", n = 

325-6". It is desired to make -j- =0.25 and in column 10 we find o 25 opposite 
— '—= 16. Hence i = 0.57" and .5= loX 0.57 = 5 7°" andi:= "^^ =o.S". 

CHAPTER XII. 
CRANKS. 

? .65. 

Various Kinds of Cranks. 

Cranks are these forms of simple levers which are so arranged 
that they may, together with their various connections, make 
entire and repeated revolutions about an axis. These may be 
divided into the following four classes : 

1. Single Overhung Cranks. 

2. Return Cranks. 

3. Double Cranks, or Cranked Axles. 

4. Eccentrics. 

These will be briefly treated in succession. 

I 166. 

Single; Wrought Iron Cranks. 

These cranks may be proportioned according to the rules 
given for simple levei-s and rocker arms {\ i^(jeiseq). Fig. 
462 shows the usual form; the arm tapers to two- thirds its base 
dimensions both ways, and is made slightly convex on the back. 





Fig. 463. 



The crank-pin is forced or driven in, and secured with a cap 
bolt. Fig. 463 shows a crank forged in one piece. In this case 
tue width of the arm at the base is determined by the necessary 
amount of shoulder on the shaft. The proportions of the pin 
are obtained from the rules in ^ 159. 

? 167. 

Graphostatic Calculations for Single Overhung 
Cr.\nk. 

The crank is such an important detail of machine construc- 
tion that it demands a most careful discussion, hence a grapho- 
static investigation of the stresses in it is here given. 

The Crank r^.t'/t'.— Having calculated d and /, draw the 
skeleton diagram of the crank, that is, the neutral axis 
A B C D E, Fig. 464, in which B C represents the axis of the 
crank arm, which in this case lies normal to the axis of the 
shaft, and is placed in its proportional distance from the centre 
of the crank-pin A, and from the bearing D. Then lay oif the 
force /"from a, normal to E a, choose the pole O of the force 
polygon (this being best placed upon aline passing through 
the end of /'and parallel to the axis E a), draw the ray a d O, 
and line d E, also the ra)' O P^ parallel \.o d E \ then a d E will 
represent the cord polygon for the bending which P produces 
upon the axle a C E, and P P^ represents the force upon the 
journal E, and /'i a the force upon the journal D. Also make 
a /^ equal to the crank radius 7?, draw F G, and this latter will 
Tse the twisting moment I J 140) which /'exerts upon the axis. 
This moment Jlfj may be combined with the bending moment 
Jifb, to give for each point an ideal bending moment. 

Mi = f yi/s + f '^M,-' + Mi (see ? 45), 

from which the polygon curve c' d' e' and surface of moments 
Cc' d' e' E are obtained. From the latter, in combination with 



the pin diameter d, and ordinate t of the base of the pin, the 
diameter of the shaft may be obtained according to formula 

(124). 

The Crank Arm. — Prolong E a to fl„, and transfer the cord 
polygon Dad to the base line B C, that is, make the angle 
a„ B C= the angle £> a d, and then will B Uo C be, with hori- 
zontal ordinates, the surface of moments for the bending of the 




crank arm due to the force P. Also make C Co^^ B bo= C C, 
then will the horizontal ordinates of the torsion rectangle 
B ba Co C be 'the moments with which /"acts to twist the crank 
arm about the axis B C. This moment may again be combined 
with the bending moment to give an ideal moment as before ; 
{i7o a' = -| rto C, draw B a' , make at any point //, the space 
H i = \ B ba, and make H li = ha h' -\- h' i) which gives the 
surface of moments B b' h F C for the crank arm. From this 
and from the diameter d and ordinate t, we can construct the 
conoidal form of the arm / K T /)/, according to formula (124). 
From this, again, the profile S T U ^' oi axi arm of rectangular 
section may be derived, the width h being assumed for any 
point and the corresponding thickness b obtained from the value 
y of the conoid, according to the formula : 



b_ 

y 



0.6 



iM 



(156) 



in using which, the second table 01 numbers at the close of this 
work will be found useful. If the position of the axis B Cdoes 
not give satisfactory results, the operation must be repeated 
with a better relation of parts. By proceeding in this manner 
the dimensions of a crank and axle may be so determined that 
the}' will be equal in strength to the pin upon which the power 
is exerted. 

In the preceding diagram the crank arm was taken as normal 
to the a.'vle. A slight inclination may be neglected, but if the 




Fig. 46.5. 

angle is greater, as shown in Fig. 460, it should be so considered 
in the diagram. The procedure is then as follows (Fig. 465): 
The diagram for the crank shape is constructed as before, the 
portion under a b being used only for the shank A B of the 
crank-pin, and the portion under C E being combined as before 
with the torsion moment, to obtain the surface of moments 
Cc' d' e' E. 

The crank arm is again subjected to bending and twisting; 
the lever arm is now B' C, A B' being made normal to the axis 



THE CONSTRUCTOR. 



105 



B C oi the crank arm, the bending polygon being a portion of 
the triangle C B' C , in which the angle at B' is equal to the 
angle d a D. The twisting force acts with a lever arm A B' ; 
its moment is obtained b}' drawing an ordinate at a' normal to 
B C, B' a' being taken equal to B' A. The combination of 
moments gives the surface of moments B b" c" C in same 
manner, and of the same use as in the preceding case. 

I 16S. 

Cast Iron" Cranks. 

The crank-pin is sometimes made spherical instead of cj-lin- 
drical ; such a one is .shown in Fig. 466 on a cast iron crank. 
The sphere will be of suitable diameter if described from the 
middle of a normally proportioned overhung crank-piu without 
making allowance for shoulder. The crank-pin is secured by 
cold rivetting the end in place, an excellent method and one 
often used. The I formed section can be proportioned by the 
iise of the table in \ 164. When li is taken as equal to the hub 



-A- 1 54 





Fig. 466. 

diameter, the cross section sometimes works out too light to be 
suitable for casting, and in such cases it must be increased 
according to judgment. Sometimes cast iron cranks are made 
simply by laying out the proper hubs for the shaft and crank- 
pin, and then joining them by an arm of rectangular section. 

If it is desired to employ the graphostatic method, the dimen- 
sions may first be determined for a wrought iron crank of rec- 
tangular section, and then doubling the depth (see | 162) for 
cast iron, and obtaining the proportions for I formed section 
according to \ 164. 

I 169. 

The Return- Crank. 

A return crank is one which is formeil upon the pin of an 
ordinar}' overhung crank, returning back toward and having 
rotation about the. same crank shaft as the main crank. Fig. 
467 shows a wrought iron return crank otherwise similar in con- 
struction to the one shown in Fig. 463. Frequently the return 
arm is on the same line as the main crank, as shown in the 
illustration, but in many cases it is different! 5' placed. The arm 
and pin of the return crank are similar in shape and propor- 
tions to an ordinary overhung crank. The arm of the main 
crank demands no especial consideration, when, as is usually 
the case, there is but little pressure on the pin of the return 
•crank. The main crank-pin must be determined separate!}'. It 
is subjected both to bending and to torsion. For this purpose 
the formula (154) are to be used, remembering that when the 
return crank is driven by the main crank the moment of the 
return crank is greatest in the middle of the main crank-pin. 

I 170. 
Graphostatic C.alcui^.ation op the Return Cr.ank. 

The graphostatic diagram for a return crank, with both main 
and return crank inclined to the axis of the crank shaft, is 
shown in Fig. 46S. The skeleton ABCDEFGHIis first 
drawn, the dimensions A B C E and F G being taken to cor- 
respond with those chosen to meet the requirements of the 
cranks under consideration. The pressure i upon the return 
crank-pin is here taken as opposed to the pressure 2 upon the 
main crank -pin. 

Force polygon. — After choosing a scale for the measurement 
of the forces, the force polygon (on the right) can be drawn. 



The line o to i, measured upward, represents the pressure on 
the return crank-pin ; O is the pole chosen on a horizontal line 
drawn through o, and the line i to 2 represents the pressure on 
the main crank-pin, measured downwards. Draw the rays o O, 
I O, 2 O, also draw the line a d' parallel to i O, until it inter- 
sects at d' the line dropped from D (the line of direction of the 



i<V4d>! 




Fig. 467. 

force 2) ; draw d' g parallel to 2 C until it intersects a perpen- 
dicular through C, the line of the force 3, which we know acts 
upward, but the magnitude of which is 3'et undetermined. In 
order to determine it, as well as the fourth force which acts at 
H, join g with //, giving H a as the closing line which is hori- 
zontal because we have chosen the pole C on a horizontal 
through o. Now draw in the force pohgon O 3 parallel to J{ g., 
then the line 2 to 3 is the third force acting at G upward, and 
the line 3 to o gives the downward force at U. Hence we have 
the figure a d' kg H as the cord polygon of the system of forces. 
At k is a zero point (see \ 132) and for convenience in showing 
the figure it is preferable to turn the triangle /■ g Ff over to the 
position (4 _^' //. The cord pol3'gon thus found will be of ser- 
vice in constructing the surface of moments, as will be seen 



.4& 



/»'' 




Fig. 46S. 

later. For the determination of the shank A B draw from A 
on the pressure i the triangle a b b', whose ordinates will serve 
to determine its dimensions. 

Crank-Fin C F D. — This is subject to bending, as shown by 
the surface of moments c d' e, and to twisting by the force I 
acting as a lever arm r^ C c — B b. In order to determine the 
twisting moment, take a I = r, and draw the ordinate / /', this 
latter will then be the desired moment, and the corresponding 
surface a rectangle on c e. Combining this, as before, with the 
trapezoid cd' e gives the surface of moments c c' d" e' e. Should 
it occur that the onlj' pressure acting is that upon the return 



io6 



THE CONSTRUTOR. 



crank-pin, the surface will be modified as follows : prolong the 
line a d' to in' , and taking this bending polygon, obtain the 
corresponding surface of moments c' d" e, from which the 
crank pin C D E can be proportioned. The minimum length / 
of the crank-pin must be that due to the pressure 2, as given 
before, for overhung journals. 

Axle F G H I. — This is subjected to bending according to 
the polygon Ffg' H, and also to torsion by the moment of the 
force 2 less that of the force i. In order to find the first, we 
choose in the force polygon a second pole O' , upon a horizontal 
passiug through the starting point of the force 2, returning the 
same pole distance. Draw 2 O' and make d g" parallel to it, 
make d 11^ C c = R, and we have in the ordinate 11 n' the de- 
sired twisting moment. Make the abscissa of the ordinate at 
a' ^ A a^= R — r, and this ordinate will then be the moment 
with which the force i twists the arm backward. Taking this 
from n n' gives the height Ff of the torsion rectangle 
F I i' f which we may combine with the bending surface in 
the manner already given, and thus obtain the surface of mo- 
ments Ff"g" h" i" I. Should the case occur in which. the 
force I becomes zero, as is the case at some points in steam 
engines when the return crank operates the valve motion, we 
have for a bending surface F fog" //, and for a torsion surface 
F Fo i I, which gives a surface of greater ordinates to be used. 
Such a case is given in unl ttered dotted outline shown upon 
the base F I. It is assumed that the portion H I is subjected 
only to the action of a torsion couple, heuce the polygon there 
becomes a rectangle. 

Return Crank Arm B C. — This is subjected to torsion by the 
force I, with an arm A A^ perpendicular to C B prolonged (its 
moment being equal to the ordinate at a„ ), and to bending by 
the arm A^ C, whose polygon is a triangle on C A^ and angle at 
Ao equal to I a a^. The reduced surface is shown 2X C B Co c". 

Main Crank Ann E F. — This is subjected to bending for- 
wards with a moment surface Do F F" , the angle at Do being 
equal to edg'', and to forward twisting with an arm D Do, 
which is perpendicular to FF prolonged; it is also subjected 
to backward bending by the force i, with a surface EoFF', 
and backward twisting by the arm A Fo normal to F F. The 
combined bending moments give the surface E d, Co F'" F, and 
the combined twisting moments the rectangular shown upon 
F F, the combination of both resulting in the final surface 
E e"' f" F. Should the force i become zero the figure will 
be increased to that shown by the dotted lines. 



The Simple Cr.\nk Axle. 

Crank axles may be divided into simple and multiple cranks. 
A simple crank axle is shown in Fig. 469. 



to P'\i\ magnitude and also parallel in direction, and at A' is a 
normal pressure, which is iV= C? tan oc and is a maximum for 
the position A'l Zj Bl. Hence we may safely assume that the 
moments with which the crank arms and the axle are bent 
attain the maximum at the same time, and are those due to the 
force P. In the example the crank pin is at E, at B and H are: 
bearings, at ^ is a couple by which the .shai^t is subjected to 
torsion due to the force P acting with a lever arm R. This 
problem is very similar to the preceding, the portion H G 
taking the place of the return crank, with the difference that 
the force at H is variable and indeterminate, but is dependent 
upon the pressure P aX E. 

Force Polygon. — In order to make the closing line of the 
polygon horizontal, draw the line B e' to any desired point e^ 
on the normal E e' , join e' with BI ; then on any convenient 
scale draw the force P, from O, in the diagram on the right, and 
make o O parallel to H c' , i O parallel to B e' and O 2 normal 
to P. Then the distance i to 2 is the upward force /", acting at 
B, and 2 to o the force P^ at H, O.^ being the pole distance. 




K, K 



Fig. ^^o. 




Fig. 469. 



The analytical discussion of such a crank axle is such a com- 
plicated matter, and the practical results are so readily obtained 
with all needful accuracy by the graphostatic method, that the 
latter is only given here. In Fig. 470 is shown a skeleton dia- 
gram ABCDEFGHci a crank axle with both arms in- 
clined. 

If we make the value of the force P, which acts upon the 
crank pin, equal to Q when it acts in the direction K M, it will 

be equal to — =^— when the connecting rod is in any inclined 

position K L\ CA. being the angle of the rod with the axis 
KM. For a constant force O the pressure P will be a maxi- 
mum when K L acts normal to L M, and this is so nearly the 
same as the value for the vertical position ML, of the crank, or 
Q . 

■ — , that this latter may be taken for the graphical exami- 

COS C\ I 

nation without a closer determination. The force at M is equal 



Axle S/iank H G. — This is subjected to bending by the force 
P^ at H. The triangle H G g is the surface of moments, and 
the ordinates may be used to determine the dimensions of the 
journal at H. 

Axle Sliank B C. — The surface of moments for bending is 
the triangle B C c. In addition to the bending is the twisting 
moment P R ; in order to determine this make O' i normal to 
/'and equal to O 2, and also make Eoe„ parallel to O, and 
equal to R, then o Fa is the desired moment, which laid off at 
C c' and A a' and combined with B C c in the manner already 
described, gives the surface of moments A B C c" b" a". 

Crank Pin D E F. — The surface for bending moments is the 
figure d ff e' d'. For twisting we have the force P^ at H, 
with a lever arm oi E e = R. Make H g = E e ^ R and the 
ordinate g g' is the desired moment, which transferred to 
f" d'" and combined with the preceding surface gives the 
surface dff" e" d" . The greatest ordinate e e" should be 
used if the piu is to be cylindrical. 

Crank Arm G F. — Draw E Do parallel to H D' normal to 
C D. We then have forward bending by the force P at Do ; 
backward bending by /^, acting at D' . The cord polygons for 
these are, the triangle Do C i (with C i = o Ho in the force 
polygon, where //„ /;„ ^= C Do), and D' C i\ which when 
combined give the surface C i" i'" for the bending of the arm 
D C- We also have a forward twisting from the force /*with 
the arm E D- =^ k koin the force polygon, and the moment o k 
acting backward from the force P-^ with a lever arm H D' = 
H I in the cord polygon and a moment / /'. The difference be- 
tween these moments laid off at D do and C Co and the resulting 
torsion rectangle combined with the bending triangle gives the 
surface C D 1 1\ so that all five portions of the diagram now 
have their moment surfaces determined. The method of using 
these for the determination of dimensions is the same as 
before. 



THE CONSTRCTOR. 



107 



The figures show clearly the various stresses at the respective 
portions of the crauk aud throw light upou the mauuer in which 
breakages occur. 

If both crank arms are normal to the axis, the solution is 
greatly simplified, and the diagram assumes the form given in 
Fig. 47'- In this we have again A B CD E FG H as the skel- 
eton, and at ^ a torsion couple whose moment is equal to PR. 

Force Polygon. — In this case the altitude e e' of the triangle 
Be' H is taken as the measure of the force P. B b" is made 
equal to ee' , b" O drawn parallel to e' H, and Ob made normal 
to Bb" , thus giving b" b as the force jP, at H, b B that at i?, 
and 06 is the corresponding pole distance. 

Axle Shank H G. — This is only subjected to bending, aud the 
surface of moments is H Gg. 




Fig. 471. 

Axle Shank A B C— This is subjected to bending, as indi- 
cated b}' the triangle B Cc, and also to torsion by a moment 
PJ?. Make e' O' parallel to C B and equal to the pole distance 
b O, diaw e'" p parallel to e' O' and equal in length to E e^= R, 
then e e'" is the desired twisting moment, giving for A C the 
torsion rectangle whose altitude A' a' = Bb'" = e e'". The 
combination of bending and torsion moments gives the moment 
surface A B Cc"' b' a. 

Crank Pin D E F- — This is subjected to bending according to 
the surface of moments CG g c, and to torsion by the force P, 
at H, with a lever arm R = CD = H f, and a moment ff = 
Gf" = Cf". By combining the twisting and bending mo- 
inents the surface C G g' e" c' is obtained, and for cylindrical 
crank pins the rectangle of a height G g" = Cc" = e e" is to 
be substituted for the irregular outline. 

Crank Ann F . G. — This is subjected to bending by the force 
Pj acting at G. The surface of moments is G Fft^, the angle at 
G being equal to f Hf ; it is subject to torsion by the same 
force acting with a lever arm H G, giving a moment G g^= G h 
^= Fi. The combination of twisting and bending moments 
gives the surface FG h' i'. 

Crank Ann CD. — Here we have bending with the force P, 
and an already known moment ee"' =s Ck at C. Twisting is 
due to the moment Cc= Cl = Dl'. For the combined mo- 
meuts these give the surface C D dk'. 




Fig. 472. 

For the same given distances of E from B and H the torsion 
stresses on the crank arms are greater for arms normal to the 
axis than for inclined arms, so that in the former case heavier 
arms are required. The torsion in the crank arms grows less 
and less the nearer the points C and G approach B and H. 
which is a point to be considered in the interest of economy of 
material. It is also to be noted that the total length of crank 
axle FG H or D CB is less for inclined arms than for right- 
angled cranks. 



In many cases a crank axle is so situated that it is subjected 
to torsion at either one end or the other. In such cases the dia- 
gram should be constructed for both sets of conditions, and laid 
upon each other, the greater value in all cases being taken. Of 
course, care must be taken to use the same pole distance and 
same scale for mearuring forces in both cases. An example of 
such a case is found in the paddle engines made by Penn, with 
oscillating cylinders, the air pump being worked from the mid- 
dle of the crank pin. The conditions in this case are somewhat 
different from the preceding, and may be examined with the 
help of the following diagram (Fig. 472) ; 

Here we have the skeleton ABCDEFGH, and not taking 
into account the force at E, the force couple gives by means of 
the cord and force polygon the moment values B b=: Cc = Gg 
= Hh, from which the following results are obtained : 

Axle Shank A B C. — Pure torsion, which, converted into an 
equivalent bending moment, gives B b' = C c' =-.-\^Bb (see IV., 
§ 16, when Md = O}. 

Axle Shank G H. — This is the same as the preceding, and 
Hh'=Gg'=Cc'. 

Crank Pin DEE. — We have here the same twisting moment 
as in the axle shanks Dd = Ff^ S b and Dd„^ Ff„-=B b'. 

Crank Ann CD. — We have in this portion a bending moment 
of the magnitude Cc" ^ Dd' ^ Cc, of w-hich the plane stands 
normal to the plane of the surface of the crank arm. The sur- 
face of moments is in this case equal to a rectangle of the height 
Bb = Cc. 

Cfank Arm EG. — In this case we have both torsion and 
bending. The couple is decomposed at G into two parts, one 
acting normal to the axis of the crank arm, and the other in the 
direction of the arm. The first gives the torsion rectangle 
G Ff" g", the latter the bending rectangle E Gii' , which com- 
bined give the moment surface F G g"' /'", in which we again 



have pq = lG /, /> r = | G g" , pt=Gi 



--qs+qr. 



Thus far we have proceeded as though there were no force 
acting at E. When such exists, however, first determine the 
bending and twisting moments as shown in Fig. 472, add or 
subtract, according to direction, the twisting moments, taking 
into account the position of the planes of bending action, and 
finally combine the bending and twisting moments so found, 
according to the method of Case IV., 'i i5. The amount of 
work which this investigation requires of the drawing-room of 
any machine-shop is small compared with the importance of a 
thorough determination of all the stresses which act upon such 
a piece of work as a crank shaft forging. 




Fig. 473- 

J 172. 
MuT,Tipi.E Crank Shafts, Locomotive Axi,es. 

One of the most important forms of crank axles made of 
wrought iron or steel is that used for locomotive engines. As 
an example of this subject, the crank axle for an inside con- 
nected locomotive is given in Fig. 473. In drawing the diagram 
of moments it is necessary to take into account tne diameter of 
the driving-wheels, as will be shown in Fig. 474. Cj and C, are 
centres of the steam cylinders. Ay and A., are the journals, aud 
By Dy and B., D^ are tlie hubs of the respective driving-wheels. 
The cranks at Cy and C, are placed at right angles with each 
other, taking the position which the axle shows in Fig. 473. An 
inspection of the figure shows three distinct loads acting upon 
the axle : l, the pressure in the vertical plane due to the weight 
of the locomotive and to the lateral action upon the wheel 



io8 



THE CONSTRUCTOR. 



flanges ; 2, the horizontal pressure of the piston against the 
crank C, opposed by a corresponding adhesion at the circumfer- 
ence of the driving-wheels ; 3, the oblique pressure of the con- 
necting rod acting upon the crank C^. Other small pressures, 
such as those due to the eccentrics, etc., may be neglected. 



H So 




Fig. 474. 

Forces and TiToments in the Vertical Plane. — Fig. 474. From 
the point Sf, of the height of the centre of gravity of the loco- 
motive laj' off the force Q, to represent that portion of the 
weight which is borne by the axle under consideration. The 
oscillations and action of centrifugal force upon curves also 
produces a horizontal force H, which may be taken as equal to 
0.4 Q. The resultant R of the two forces O and H is the load 
upon the axle. This may be decomposed into the pressures P^ 
and P, upon the journal at A-^ and A,_. and into the pressures 
(2i and O., upon the wheels at E-^ and E.^, which pressures, with 
their reactions, produce the stresses on the axle. The forces Q^ 
and O, can be decomposed into two others referred to the wheel 
hubs i?i Z?! and /?, D^- This gives six vertical pressures acting 
to bend the axle, viz. : i, 2, 3 and 4 acting downward at D^, A-^, 
A2 and D.,, and 5 and 6 acting upward at B., and B^. From 
these forces, by choosing anj' desired pole distance, the force 
polygon E, 4, O may be constructed, and also the cord poly- 
gon or surface of moments d^ a^ a„ d^_ b„ b^, and this surface gives 
hy its ordinates the proportional bending moments in the verti- 
cal plane for each point in the axle ; this entire surface is desig- 
nated by the letter /'. 




Fig. 475- 

Forces and Moments in the Horizontal Plane. — Fig. 475. As 
already shown in a preceding paragraph, the pressure P ovi the 
crank pin for the position L yl/of the crank is somewhat greater 
than the pressure P^ on the piston ; its moment of rotation about 



the shaft is 



P. 



. R cos n, which = P^ R, so that upon the as- 



cos a 

sumption that the wheel on the left slips on the rail, the other 
one must oppose a resistance whose moment equals Pf, R and 

the frictional resistance 3 at £'2 = /o — Combining this force 



3 at E., and also the force 4 =: P^, and the resistances i and 2 at 
the journals, we are enabled to construct the force polygon 
.^j 2 (9 and the corresponding cord polygon // for the horizon- 
tal forces, as shown in the light sectional portion of the diagram. 
The forces 1 and 2 are found bv taking tlie position of the re- 
sultant of the two forces 3 and 4, as shown in the figure, and 
decomposing their sum into the portions which would go re- 
spectively to x4^ and A.,, as shown by the construction given in 
the dotted lines. 

Forces and IMoments in the Inclined Plane 0/ the Connecting' 
Rod. — The force Q = S acts at (Ti, making an anale with the 
horizontal equal to 3/ K L. As shown in the illustration, this 
maj' be decomposed into the two opposing forces 6 and 7 at A^ 
and A.,, and by taking the same pole distance as before to con- 
struct the force polygon we obtain the cord polj'gon 5, shown 
by the dark section lining, and giving the surface of moments 
for bending in the inclined plane of the connecting rod. 




Fig. 476. 

Combination of the Three Preceding Cord Polygons for Bend- 
ing of the Axle. — Fig. 476. Since the three preceding sets of 
forces are acting at the same time to produce bending in the 
axle, it is necessary to combine the diagrams in "order to obtain 
the final result. For this purpose we can treat the respective 
ordinates in the same manner as if they were forces, as in \ 44. 
Taking the successive points upon the axle, we construct the 
corresponding ordinate polygons, whose closing lines give the 
resulting moment both in direction and magnitude. One of 
these ordinate polygons is shown in the upper portion of Fig. 
474, to the left: it belongs to the point Q. The vertical ordi- 
nate /'in this case acts upward, the horizontal ordinate //^con- 
tinues toward the left, and the inclined ordinate 5' also continues 
to the left, thus giving the resultant T as the line joining the 
origin of V with the termination of 6". We thus obtain for the 
entire axle the surface of moments ZJ.j /?i flj f , fj ^22 ^^27 which 
gives the proportion of bending stresses of the axle, as distin- 
guished from those of the crank arms. 

The Torsional ^Foments for the Axle. — The position of crank 
described above and selected for this investigation gives a tor- 
sional moment only upon the crank to the left, and also one of 
the magnitude PR upon the axle extending to the point D„. If 
both cranks stand at an angle of 45° with the horizontal, there 
will be produced in both end shanks Cj D^ and C^ D., moments 
equal to \/ 2 PR, or about 1.4 PR. Under these circumstances 
the moments at the ends become Z?, (// ^ /?, a'./, while in the 
body of the shaft (T, C, we have the moment C^ <"/ ^ C", c' = PR, 
always keeping the scale of forces and the pole distance the 
same in all of the diagrams. It must be remembered that in 
this position of the cranks the bending moments are somewhat 
different from those shown in the preceding diagrams. 

Combination of Bending and Tivisting liloments. — The 
bending and twisting moments can now be combined accor- 
ding to the formula of §45, and thus the surface of moments 
D^D^d^b^ . . . . dr," obtained, b}^ the help of which the 
shanks C, D^ and C, £>., and body of the axle C, C, can be pro- 
portioned, after the diameter for any one of the ordinates, as, 
for example, that at B^ 5,, has been determined. The half of 
the diagram which gives the greatest ordinates should be used 
for both halves of the axle. 

Crank Pin at C^- — The two crank pins are treated separately 
in Figs. 477 and 47S, since the moments can be laid out more 
conveniently in that way. For the pin EG at Cj we have, iu 
addition to the bending moments obtained from Fig. 476, and 
shown by the surface EG c^, the combined forces on the left, up 
to the point E, acting to twist the pin. The resultant of these 
forces is j^et to be found. The vertical forces are those shown 
at I, 2 and 6 of Fig. 474, their algebraic sum being shown at /, 
in Fig. 477. The horizontal force acting backwards is //, repre- 
sented as I, iu Fig. 475. The inclined force acting downwards 



777^ CONSTRUCTOR. 



109 



and back'n-ards, shown at ///, corresponds to the force 6, of 
Fig. 475. The closing line (not shown) from /// to C^ would 
give the resultant, and its horizontal component //•''acts to twist 
the crank pin FG, with a lever arm EF-^R. In the force 
polygon (above, on the left) we take <z O to be the pole distance, 
as before ; lay off IV downward from O, draw alV e, make 
af^= R ; then will fe, perpendicular from f, be the twisting 
moment Ff. Combining this with the surface of bending mo- 
ments F G (Tj, we obtain the final surface FG c/. 




ba"*' bj 



Fig. 477- 



Crank Arm E F. — The ordinate polygon V^ H^ S^ T (on the 
left) is constructed for the point E. The horizontal component 
/Z[ of the resultant T^ acts to twist the arm E F, Fd=^h^ ; the 
vertical component 'c\ acts to produce a bending of the arm in 
the plane of the diagram, Fb^z\ ; also the force //'acting at 
E tends to bend the arm normal to the plane of the diagram, 
with a moment 5 0, = F dy at F. The combination of the bend- 
ing moments gives the surface E Fb' b", which, with the tor- 
sion rectangle E Fd, gives the final surface E Fb'". 

Crank Arm G H. — The ordinate polygon / ', H^ S, T^ is con'- 
structed for the point H. The horizontal component //, acts to 
twist the arm G H, M d^ = /;„ ; the vertical component v, shows 
the bending in the plane of the diagram, G b^^=v.,\ also, the 
force /-"bends the arm normal to the plane of the diagram with 
a moment PR =f/i, of the force polygon above, on the left, in 
which 0,0- = /■,«/"= /?. Again, make (5/ 53 =/'//. The combi- 
nation of the bending moments gives the surface G H b/ b.,", 
and the combined bending and twisting moments give the final 
surface G Hb'". 




Fig. 478. 

Crank Pin K L. — Fig. 47S. This crank pin is subjected to 
the bending moments which act between 7!/ and_/, and indicated 
by the surface K L r,, obtained from Fig. 476. The collected 
forces which act on the left of Q tend to twist the pin. The 
resultant of the forces 3, 4 and 5, Fig. 474, shown at V \\\ Fig. 
476, acts downward, the resultant (difference) of the forces 2 
and 3, Fig. 475, and shown at VI, acts horizontally backward, _ 
and the force 7, of Fig. 475, shown at VII, acts inclined back- 
wards. The vertical component of the force polygon V, VI, 
VII, acts to produce twisting at y>/, remembering that the crank 
J IC is taken in the horizontal position. The moment of this 
vertical component has the magnitude kk'. Also we have act- 
ing to twist the pin the couple shown on the left (as discussed 
in connection with Fig. 472) with a moment already determined 
and shown at C-^c/ in Fig. 476, and here laid off at K k, from 
which, since the previously determined twistiug moment k k' 
acts in the opposite direction, we must subtract kk', giving 
finally for the crank pin K L the twisting moment Kk', which, 
when combined with the bending moment, gives the surface 
KLc^'. 

Crank Ann J K. — This is subjected to twisting by the moment 
A'(/=the vertical component v„ of the ordinate polygon 
V^H^S-^T^. For bending in the vertical plane we have the 
moment Kl^Kk, as already shown in Fig. 472 ; also in the 
same manner and direction by the vertical component of the 
forces V, VI and VII with the moments b b^ at K (see the dia- 



gram of these moments in the upper left portion of Fig. 477). 
It is subject to bending in the horizontal plane by the horizon- 
tal component /i^ of the ordinate polygon, the moment being 
b b^. The combination of bending moments gives the surface 
/ A'b/ b./, and the final combination with the twisting moment 
Kd gives the surface / A' i.^". 

Crank Ar»i L 3J. — The twisting moment is L d, = the verti- 
cal component i\ of the ordinate polygon for the point M. The 
bending moment L b.i = K k, also b, b^ due to the vertical force 
at I\I, and also the bending moment i-, 65 = the horizontal com- 
ponent h^ of the ordinate polygon. The combination of bend- 
ing moments gives the surface ill Lb/, and the final combina- 
tion with the twisting moment gives the surface ML b/' . 

Of the four crank arms, J K is subjected to the greatest stress 
at the pin, and G H at the axle. In practice, therefore, the 
surfaces y A' A/' and C/Zi^,"' should be drawn upon each other 
and the greatest ordinate used. The resulting dimensions, with 
possibly slight modifications, should then be used for all four 
arms. 

Although the construction of such a graphostat'c diagram of 
moments involves some labor, the result is most satisfactory, 
since by assuming a stress of say 3 the modulus of working 
stress (about 17,500 lbs. for wrought iron, 25,000 lbs. for steel) 
the design can be properly proportioned without further care. 

The calculations for locomotive axles with outside cranks is 
similar to the preceding, although the diagrams are necessarily 
somewhat different, although laid out in the same general 
manner. 




Fig. 479. 



Fig. 480. 



i 173- 
Hand Cranks. 



The chief peculiarity in a hand crank lies in the adaptation 
of the crank pin to be operated by hand. In Fig. 479 is shown 
a crank for two men, and in Fig. 4S0 for one man. The dimen- 
sions for the parts indicated by the letters are as follows : 
For 2 men. For i man. 



/?= 14" to \W' 


12''' to W 


I' = 16" to 19'' 


11" to 13'' 


D~ I J" to If" 


ii"to I*' 



The other dimensions figured in the illustrations are in milli- 
metres. When placed at opposite ends of the same shaft, hand 
cranks should be set at 120° with each other. 




Fig. 4S1. 



Fig. 482. Fig. 483. Fig. 4S4. 



? 174. 

Eccentrics. 

An eccentric is nothing more than a ciank in which (if the 
crank arm is R and the shaft diameter D\ the crank pin diam- 
eter d' is made so great that it exceeds /? -(- 2 j?, or is greater 
than the shaft and twice the throw. The simpler forms of eccen- 
tric construction are shown in the illustrations. The most prac- 



no 



THE CONSTRUCTOR. 



tical of these is that shown in Fig. 4S3, the flanges on the strap, 
as shown in the section, serving to retain the oil and insure good 
lubrication. 

The breadth of the eccentric (properl)- the length of pin /) is 
the same as that of the equivalent overhung journal subjected 
to the same pressure ; for the depth of flange a we have 

a = 1.5 ^ ^ 0.07/ -j- 0.2 (157) 

from which the other dimensions can be determined as in the 
illustrations. 

For some forms of shafts with multiple cranks or other ob- 
structions the eccentrics cannot be made as shown above, but 
must be in halves, bolted together. 



CHAPTER XIII. 

COMBINED LEVERS. 

I 175- 

Various Kinds of Combined Levers. 

Two simple levers with the same hub form what is termed a 
Combined Lever. AVheu both arms have a common centre line 
they form a Beam, or so-called Walking Beam ; and when they 
form an angle with each other they are called an Angle Beam, 



J,=OB 




Fig. 4S5. 

or frequently a Bell Crank. The pressure {?, upon the axle of 
an angle lever A O B, Fig. 4S5, is determined by the relatio- 

Q = V/'i- + P.^ — 2 P, P, cos a 

if P^ is the force acting at A, and P„ that at B, both acting at 
right angles to their respective arms"; a being the angle between 
the arms. This may be shown graphically by making P^ = OB 
and P.^ = A, when Q will = A B, the, third side of the tri- 
angle. If the forces /*, and P., do not act at right angles to the 
arms, the triangle must be constructed bv drawing lines from 
O, normal to the directions of the forces. 

The variety of combined levers is very great, and only a few 
of the principal forms are here given. 




Fig. 4S6, 

?i76. 
Walking Beams. 

One of the principal forms of combined levers is the walking 
beam, for use in some forms of steam engine. These are usually 
made of cast iron, with journals and pins similar to those given 
in Fig. 456 ; and other forms of journals are also shown in the 
following figures. 

Fig. 4S6 a shows an ornamented beam-end, with the pin keyed 
fast Fig. 4S6 b shows a beam-end with a bored cross-head and 



pins cornbined, fitted on the turned end of the beam and secured 
by the pinned collar shown. This construction requires careful 
fitting, and is somewhat expensive. 

Fig. 4S7 a. This is a fork journal ; the fit is made with a very 
slight taper, secured by cap bolt and large washer at one end. 
The pin is kept from turning by a projection under the head, 
let into the boss on the beam. 




Fig. 487. 



Fig. 487 b. This is a spherical bearing with its shank driven 
into the end of the beam and keyed fast, this form .giving great 
freedom of motion to the connecting rod.. 

The diameters of pins are determined as already given in I 90. 
The load is to be considered as acting continuously or intermit- 
tently, according as the engine is single or double acting. 

Fig, 4SS shows a form of beam which has been extensively 
used. In order to secure lateral stiffness, the beam centre should 
not be made too short. A good proportion is that given in the 
figure, in which the distance between centres of bearings is 
.made equal io 6d + ^\ A. The distance between centres of 
journals for the ends of the beam is made from 4.6 a'j to 5.5 d^ ; 




Fig. 4SS. 



<^i being the journal diameter, as shown. The depth h of beam 
m the middle must not be made less than 



k = Ad + 



A 



(15S) 



in which d is the diameter of the beam centre, and A the half 
length of the beam. If the two arms are of unequal length 
their mean should be taken.* 

The curved outline of such beams is drawn according to the 
methods given in g 142, starting from the crown of the beam to 
the hub for the pins at the ends. The ribs in the middle of the 
beam are given the same thickness, c, as the flange at the edges, 
and the breadth of flange is shown in the plan at B (see I 163)' 

Another form of beam is shown in Fig. 4S9. This is made 
double, and in such case each half is calculated separately. In 
Fig. 490 IS shown a section of such a double beam in which the 
parts are somewhat widely separated. The two plates are firmly 
bolted together, the bolts passing through tubular sti ts, as 
.shown, and the parallel motion rods are hung between the two 
parts of the beam. 



* In the United states much greater depth is given to beams of this sort 
sometimes 2 to 2}^ times that given by the formula. Skeleton beams w°th 
cast-irou centres and wrought-iron bands are also much used. 



777^ CONSTRUCTOR. 



Ill 



A beam of somewhat unusual form is shown in Fig 491, being 
a poition of the hydraulic riveting machine of Mackay & Mc- 
George, built by Rigg." The beam centre is at A, the rivet die 
at B, the hydraulic pressure is exerted by small and large cylin- 
ders at D and C respectively. The water pressure is taken from 
au accumulator and discharged into an outlet pipe placed some- 
what higher than D. By lueaus of a suitably arranged valve 



bolts. Fig. 494 shows the form used on American locomotives. 
The example is from a passenger engine, and extends between 




Fig 4S9. 



gear the high pressure water is first exerted upon the small cj'l- 
inder, and water from the discharge pipe delivered to the large 
cylinder, thus closing the die upon the rivet at B. Then the 
high pressure water is also delivered to the large cylinder, 
making a still greater pressure upon the rivet, with practically 




Fig. 490. 

no expenditure of water, as that cylinder is already filled. The 
pressure upon the rivet is 60 tons. The beam is made of a sec- 
tion of uniform resistance (see 2 9). At .£" is a short shear for 
cutting beams, angle iron. etc. The distance B Cis 12 feet. 
Wrought iron beams are not uncommon, and for moderate 




Fig. 492. 

loads and dimensions are conveniently made in the double form, 
as shown in Fig. 492. The depth /; in the middle may be taken 
at 0.8 times the value given b}' formula (158). For larger beams' 
of wrought iron, the girder form shown in Fig. 491 is to be pre- 
ferred. 




Fig. 493. 

Another form of beam is the equalizing lever, used to distrib- 
ute the weight among the springs (see Figs. 102 and 103, ? 41). 
In Fig. 493 is shown a lever of ViTought iron for a heavy engin-^ 
{the Prussian standard freight engine). The length A B is iiSo 
mm. = 462", and the connections at A, O and B are made with 

* See Engineering, March, 1S75, p. £23. 




Fig. 494. 

the springs of the driving-wheels, being jl feet long. At O, A 
and B are half journals, and the connections at A and B are 
not rigid. The bearings are not on a straight line, as in the 
German form, but the variation is trifling. 




Fig. 491. 

Sc.il^E BE.IMS. 
In scale beams the bearings are usually made upon knife 
edges (see ? 95), generally with an angle of 60°. A special form 
is here given, Fig. 495, which may serve as an example, showing 
the main supporting beam of a bridge-scale, in triangular form. 
In the construction a, the main bearings are at OO; the bear- 
ings A A form a double journal analogous to Fig. 476 ; at j? is 
the end journal, here set in a cast-iron head. In the form shown 
at b, we have two separate bearings at O O, the parts being held 
together by a bolt C* 



* For similar examples see E. Brauer's ' 
Construction), Weimar, Voss, iSSo. 



Konstruktion der Waage " (Scale 



112 



THE CONSTRUCTOR. 



Scale beams should show very little deflection under their 
load. They are therefore made very deep in proportion to their 
total section, aud the stresses taken at 4250, S500 and 14,220 lbs. 
respectively for cast-iron, wrought-iron and steel. 




Fig. 495- 



CHAPTER XIV. 

CONNECTING RODS. 
I 178. 

Various Parts of Connecting Rods. 

Connecting rods are used in various forms for transmitting 
the motion of various reciprocating parts of machines to levers, 
beams or cranks, or vice versa. It is necessary to consider sepa- 
rately the ends or heads which contain the bearings for the 
crank and cross-head pins, from the body of the rod. The 
dimensions and proportions of the ends are governed, to a 
greater or less extent, by the dimensions of the bearings, the 
latter being either forked, overhung or necked, and their size 
determined by the pressure to which they are subjected. 

I "79- 
Connections for Overhung Crank Pins. 

The strap and key connectiou shown in Fig. 496 is widely 
used. The boxes are surrounded and drawn together by the 




liHil |ii'pf?-':'-I|| 




"iS?..^... 




Fig. 496. 



strap and key, and by driving up the latter they may be closed 
together to take up wear. In determining the dimensions, the 
boxes and their surrounding parts -vfrill be considered separatelj-, 



as in the case with other bearings. The unit or modulus for the 
boxes is 

e ^= 0.0"] d -\- o.wW (159) 

being the same as used or other bearings, d being the diameter 
of pin. 





Fig. 497- 

Fig. 497 shows two views of the brasses, the dimensions of the 
other parts being based on the following modulus : 

rt'i = 0.0267 \/P+ 0.2'' (160) 

The breadth b may be made equal to 0.8 ^j, or if the length of 
the journal is made equal to its diameter b becomes = d — • 2 <?. 

Example : If P= 7920 pounds alternating load, we have from (93) d= 2%"^ 
I also = 2j's", and according to (166) 

d\= o 0267 \/7920 -1-0 1" 

= (o 0267 X SS.g) -I- 0.2" = 2.57", say if^' 

We also have e = (0.07 X 2-375) \ 0.181 = 0.3". Also b ^ I — 2e = 2.375 — 
0.6 = 1.77. Applying the value of rfi to Fig. 496, we have the thickness ot 
strap = 2.57 X 0.2 = 0.514 on the sides, and 2.57 X o 3 = 0.76 on the end ; also 
the thickness of key = 2.57 X 0.22 = 0.56", and the other dimensions in a 
similar manner. 

The key must be given much less taper when it is used with- 
out a set screw, as in the illustration, than when a set screw is 
used. In the former case a total taper of yV is used, and in the 
latter \ is safe. 




M^. 



Fig. 498. 

The boxes are best made to bear closely together instead of 
being set open, as shown in the figure, and better practice in 
this respect is shown in Figs. 499 and 500. In this case the 
boxes must be filed off to permit them to be closed up for wear. 

Au objection to the form of strap end just shown is that the 
continual keying rap of the boxes tends to shorten the rod. The 
reverse action takes place with Sharp's strap end. Fig. 498, the 
action of keying up tending to lengthen the rod. 

In Fig. 499 is shown a capped end of solid bronze, as made 
by Penn. The two halves are fitted closely together, so that 
the joint must be filed out to take up for wear, or else a num- 
ber of thin slips of copper may be inserted in the joints aud 
removed one at a time, as may be found necessary. The diam- 
eter & of the bolts must be made, so that they shall not be less, 
measured at the base of the thread, than the value given by 
formula (84). For V thread this is given by making 



! = 0.0142 y 



P 



and if square thread bolts are used they should be made slightly 
larger. The stress on the material with these sizes will then be 
between 7000 and 8000 pounds, which is not excessive. (Com- 
pare Example 2, \ 1S2.) 



THE CONSTRUCTOR. 



"3 



Tlie iiuts of these bolts are fitted -witli Perm's locking device, 
Fig. 243. For rods of large dimensions, such as are used on 
heav}' marine engines, the boxes are cored out in order to secure 
economy of material. 





Fig. 499. 



In Fig. 500 is shown a rod with a solid end, and is a very ex- 
cellent form, and with proper tools, not too expensive to con- 
struct. The boxes are made of bronze, lined with white metal 
and turned on the outside. The movable box is fitted with a 
wrought iron pressure block, which receives and transmits the 
pressure of the kej'. The boxes are provided with small pro- 
jections, which engage with corresponding recesses and prevent 
them from turning. 

The key-slot in the rod is made with semi-circular ends, 
partly because the machine which forms the slot leaves it in 
that condition, and also because this shape weakens the section 




Fig. 500. 

of the rod less. The key itself is made flat on the side which 
bears on the pressure block, in order that liners may be intro- 
duced when necessary. The key is secured by the method 
shown in Fig. 201. It will be noticed that the nut is set so deep 
in the recess that a socket wrench is required to turn it. This 
is done in order that nothing may project beyond the dotted 
clearance line. 

In Fig. 501 is shown another solid rod end, much used on 
locomotive engines. The boxes are made without flanges on 
the back, so that they can readily be removed after taking out 
the key. In this case there is no pressure block, but the box 
upon which the key acts is given instead a thickness of 3 fin- 
stead of 2 e. The method of securing the key is the same as 
before. An oil box is formed on the upper side, and covered 
with a brass lid attached by screws. The hole in the oil cup, 
shown in the plan, is fitted with a tube and wick. 

In both of these forms the tendency of the wear is to shorten 



the length of the rod, and if the reverse is desired, the key may 
be placed behind the other box. These rod ends are designed 
so that as much of the work as possible may be performed by 
the lathe, and sharp angles and corners have everywhere been, 
avoided, as they tend to weaken the material. 



d+l,6e 




Fig. 501. 

A third form of solid end is shown in Fig. 502, and is designed 
by Krauss, of Munich. It is intended to be made of steel, and 
is very compact and simple. The key is made in two parts, and 



y^-.^l.^.^>fi.;K.^r=.irf.^35g^.^jjgl^a^H 




Fig. 502. 

is combined with its own locking device. The boxes are made 
of wrought iron lined with white metal, and an oil chamber is 
formed in the one shown on the left. 




Fig. 503. 

Fig. 503 shows an end of cast iron, also made solid, and with 
the key acting to take up the wear from below, much as in the 
design of Sharp, Fig. 49S. Cast iron rods were formerly much 
used on the parallel motion connections of beam engines. 



114 



THE CONSTRUCTOR. 



I I So. 

Stub Ends for Fork Journals. 

Fork journals designed according to the method previously 
given, are made much smaller in diameter than the correspond- 
ing overhung journals. On this account the breadth b' cannot 
be determined in the same proportion to the diameter of pin d, 
as with overhung pins, as the pioportion will vary somewhat 
for various conditions. In order to take this difference into 
account, we may take for such rod ends, instead of the modulus 
^iven in (i6o), the following : 

■ . S+v>v^ '-'^^ 

in which b is the breadth corresponding to the length of the 
normal pin, and (/jits modulus, calculated according to (i6o). 
This enables us to use the proportions of all the preceding ex- 
amples for fork journals as readily as for overhung crank pins. 
The thickness of metal c in the boxes may be made the same as 
before, using in every case the actual diameter d' of the jour- 
nal. The formula (i6i) assumes the same material to be used 



\fi' /^ 




Fig. 504. 

in both cases, and gives the rod end for the fork journal 
approximately the same strength as one proportioned for a nor- 
mal pin. It is not, however, possible to make an empirical 
formula cover every case, and some examples will be found 
much heavier, such as, for example, would give a modulus of 

One of the portions whose dimensions will not bear much 
reduction is the key, since it is subject to shearing action and 
its limited surface must not be subjected to too great pressure. 



/i \b' ' d ) 



0,5d 




r*--*?- 


!-l— *, 


m^i 


1 .] 


— Eljf^ ^ 


1 "- t-~=3 ^ 


i^h 


J ■--_=%_ 


f^"! — '■■■ -- 


-^m% 




Fig. 505. 

For this reason the dimensions of the key should in no case be 
made less than those given for stub ends for overhung pins. 

Example. Given an alternating pressure ol 7920 pounds, let it be required to de- 
sign a strap end similar to Fig. .196. We have for an overhung pin according to (93), 
</:= 2^s", and for a fork journal, according to (98), d' = i-Je"-- and the length /' of 
the latter = -z d' = "iYz" . We then give the strap the proper breadth for the over- 
Jiung pin, that is, d — 2 <7, or 2.375 — o-sS = I-795- say i-fg''- ^^ then have, for the 
moduli, respectively; for the overhung pin, according to (160), d\ = 2.56", and for 



V 



Ax- = =-56 



-Ji 



i75 



the fork journal, according to {i6i), di = 2.56 

^ i/ d "^ 2.375 

3.16. From this we get for the thickness of the strap 2 16 X 0.2 = 0.432", say iV'i 
on the sides ; and at the end, 1. 16 X 0.3 = 0.64S. or nearly \\''. They may be made 
the same as for an overhung pin, giving a thickness of 2.56 X 0.22 = 0.56", or ^^5", 
and a depth of 2.56 X 0.2 = 0.512 at the small end. 



In Fig. 504 is shown a solid wrought iron end, suitable for 
forked journals as made at Seraing. In the plan shown in Fig. 
505 the journal and fork are formed in the rod end, and the 
bearing is made in the crosshead, as shown later in Fig. 540, 
§ 189. Such rod ends have been used for locomotives by Polon- 
ceau, and for marine engines by Humphrey. In these cases the 
values of b' and h, must be chosen to suit the space at the dis- 
posal of the designer in each instance. 





Fig. 506. 



Fig. 507. 



For fork bearings in which there is but little angular move- 
ment, as, for example, in valve-rod connections, the form shown 
in Fig. 506 may be used. In this case the key and block press 
upon the half bearing of the outer part of the portion B, as 
shown. Such connections are sometimes also made by using a 
flexible steel plate, as shown in Fig. 507 ; and this form may be 
called a plate link. This has been used in some forms of loco- 
motives and in the old style Laugen Gas Engine. 




Fig. 508. 

In Fig. 508 is given an end for a fork journal, such as would 
be a suitable one for the cross-head end of the rod shown in 
Fig. 500. The boxes are made cylindrical and fitted with a 
wrought-iron pressure block. The pressure of the key is trans- 
mitted to the block by a bronze intermediate, but this arrange- 
ment of key involves much clearance space. The method of 
securing the key is that shown in Fig. 200, and the whole de- 
sign is well worked out. 

Another form for the cross-head end of a rod is shown in Fig. 
509, and this is well suited to be used with the form given in 
Fig. 501, on the crank end, for locomotive use. The key is set 
up by turning the screw ; the latter can. also be secured at every 
sixth of a revolution, b)' means of the arrangement of pin aijd 
washer as shown in Fig. 237. 

I 181. 

Connections for Enlarged or Neck Journals. 

As shown in g 92, there is no definite relation between the 
diameter d' of a neck journal and the diameter d of the corre- 
sponding overhung journal ; hence it is impracticable to use the 
rules dependent upon the length of the overhung journal, which 
have been given in the discussions of return cranks, crank axles 
and eccentrics. It is, however, necessary to devise some method 
of proportioning the rod ends for such cranks, and for this pur- 
pose we may use the figures given for overhung crank pins, by 
making a modification in the modulus according to formula 
(161). In such cases we must remember to use the value of d' 
ia determining the unit e for the proportions of the boxes. 



THE CONSTRUCTOR. 



"5 



^j:fl7«^/^.- -Suppose, instead of the fork journal of the preceding example, we 
have a neck journal of a diameter d' = 43'i", and length /' =^ s'a", with a stub end 
iike that shown in Fig. 496. We have d^ = 2.56", b = i .S125", d = 2.375, We may 
make the value of ^' the same as for the corresponding overhung crank, or ^' ^ ^, 

and we then obtain from (161) d-^ = ^1 -Vf ^'"^^ = '-5^ X 1-4^4 = 3.62", 

> 2-375 
For the boxes we have e = 0.07 X 4-75 + 0.125 = 0,45'', say /g". 

In the following examples are given modern designs for rod 




Fig. 509. 

ends for neck journals, and others may be obtained by modifi- 
cations of the preceding forms. 

Fig. 510 shows a solid end connection for a spherical journal. 
ITlie sphere in this case is made 1.5 times the diameter of the 



Fig. 511. 



Fig. 512. 




Fig. 510. 

corresponding cylindrical journal, and an example of this form 

d' 
may be seen on the beam in Fig. 4875. This gives — ~ = 1.5 ; 

a 

and if, as before, we make b' = 6, we have d\ = di y i-S = 
1.225 d^. If, again, d = 2.375'', we have d' = 3.56", tfj = 2.56", 
and d\ = 2.56 X 1-225 = 3-i3''. say 35". The boxes are made 
without side flanges, so that they can be removed by backing 
out the key. The key may be arranged to be fitted above or 
below the boxes, as may be desired. When used upon locomo- 
tive engines, this form is sometimes strengthened as indicated 
by the dotted lines. 

For the connections of crank axles, return cranks and similar 
situations it is necessary to use a form of rod end which can 
be opened. The following forms are designed for this purpose, 
being made with blocks which are firmly bolted in place, but 
readily removable. 

Fig. 511 shows a form similar to Fig. 500. The block is fitted 
between two shoulders and also secured by two through bolts. 
Fig. 512 shows a design by Krauss, in the same stjde as Fig. 502, 
and used with it on a locomotive connection. The block is here 
made of bronze, and also forms one-half of the bearing ; it is 
held in place by a through bolt, which is omitted in the draw- 
ing. A cross-section is shown above, the offsets serving to keep 
the block from twisting on the bolt. The gap between the boxes 
is filled with slips of copper. The rod and bolt are both made 
of steel. 



Figs. 513 and 514 show two forms of eccentric straps, both 
intended to be made of bronze. The breadth // is equal to /, 
the length of the corresponding cast-iron journal (see | 92). 
If d = lyV'i '^i = i-^". if = * = 2.375", we have, if rf' = 15.75, 



b' = l= 2.375'', d' 



-^■^il 



5-75 



5625 



: 5.71"- The diameter, d, 



of the bolts of these eccentric straps is determined from the 
following : 

tS ^ 0.33 fi?j -(- 0.06 o'l' (162) 

in which d-^' is the modulus for a neck journal and d^ the mod- 




Fig. 513. 



Fig. 514- 




Fig. 515. 



Fig. 316. 



ulus for the corresponding overhung pin. If we take the values 
above given, rf/ =: 5.71" and d-^ = i.S", we get 

(! = 0.33 X 1-8 -1- 0.06 X S.71 = 0.9S66", say i". 

If we make d' = d and d-/ = t/,, we obtain from (162) the 
same dimensions as on a capped and bolted rod end. 



ii6 



THE CONSTRUCTOR. 



In Fig. 515 is shown a design for a cast iron strap, with bronze 
lining, although this latter may be omitted. The eccentric rod 
is secured by means of a key, and if two eccentrics are placed 
close side by side, the keys should be placed at 45° from the 
position shown. 

Fig. 516. This is a wrought-iron strap, also lined with bronze. 
In this, as in the preceding example, the joints between the two 
halves of the bronze lining are close, and those of the strap are 
open, and by filing the ends the halves may be closed together 
to provide for wear. Instead of forging the rod in one piece 
with the lower strap, it may be made with a T head and bolted 
fast, as shown by the dotted lines. 

Example The eccentric straps on the engines of the "Arizona," 6600 H. P., by 
John Elder & Co., of Glasgow, are made as in Fig. 515, but with the rods attached 
by T heads, as described above. The diameter of eccentrics d' = 54", the breadth 
I = 5", and the shaft diameter = 22 J^". 



rod is made with a forked end, and two bearings, its lateral 
stiffness is thereby increased, and m may be made as low as 4, 
If w = 20 we have for wrought iron or steel, C^ 0.0346. 

Example i.— For a wrought iron connecting rod 118.11" long, acting under 
a pressure of 31,680 pounds, taking m = 20, and C = 0.0346 we have a 

diameter D = 0.0346 J II8.Il^/3l68o = s". 

This gives the diameter in the middle ; it may be somewhat 
reduced at the ends, these latter being made of a diameter 
= 0.7 D, giving a cycloidal sinoide as in Fig. 5, formula (23). 
The ends of the rod should be worked off into the body m such 
a manner as not to make too abrupt a change of cross section. 
This becomes more important in high speed engines. In the 
case of locomotives there is sometimes a marked bending action 
upon the rod, there being a so-called "whip action" at every 
revolution of the crank, dependent upon the rotative velocity 



§182. 
Round Connecting Rods. 

The body of a connecting rod may be made of wrought iron, 
cast iron, steel, or even wood. In the latter case it is usually 
only subject to tension. 

If the rod is of circular cross section, of diameter D, and 
the force of tension be P, we have the following relations : 



Wrought Iron 



Steel 



Cast Iron 



D 



:0.014s 



VP 
D 

Vf 
D 



= 0.0117 



Oak 



^P 

D 



Vp 



= 0.0578 



(163) 



These give stresses of 5600, 9500, 2S00 and 400 pounds respec- 
tively, or above two-thirds the value given for ordinary condi- 
tions. 

These formulce ma}' also be used for short rods which are 
subjected to compression, but if the length L, of the rod is so 
great as to permit bending, the diameter must be made some- 
what greater. From an examination of case II, \ 16, and also 

I 127, we should not permit P to be greater than — i^^— , in 

v^hich/ is the moment of inertia of the cross section of the 
rod, and E the modulus of elasticity of the material employed. 
In order to determine how small P must be, or rather how 
large the co-efiicient of safety in, must be taken so that we 

shall have P = — . i-^ , there are various conditions 

m L'- 

to be considered ; the requirements being almost as varied as 
in the case of columns. 

Leaving then the value of ?«, to be subsequently determined, 

V7e have / = -^ D'' and E = 28,400 000, for wrought iron and 

&4 
steel, 14,200,000 for cast iron, and 1,562,000 for oak, and hence 
the following formula for the diameter of rod. 



Wrought Iron or Steel D = o.oi 



64 ^» 



Cast Iron 

Wood 

We have for 



s]lVP 

-^lVp 

D = 0.034 ■^'« sj -L ^ P 



D: 



; 0.0195 V ;k 



(164) 



^« 



1.5 

I. II 



3 
1.32 



4 
1.41 



10 15 
1.7S 1.97 



25 

2.24 



3° 
2.34 



40 50 60 
2.51 2.66 2.78 



If we represent the entire co-efficient of a//, \^ P by C we 
may write for the above formulce 



and may then determine values for C according to the degree 
of security required. As already stated, there is a wide variety 
of values of in to be deduced from practice. For stationary 
engines of moderate size we find in, very high, often 50 to 60. 
These however are not to be taken as standards because they 
are rarely designed for economy of material, but rather for per- 
fection of action. For medium and large stationary engines 
we find m from 5 to 25, probably averaging about 20. If the 




and the weight of the rod. This action also occurs in a lesser 
degree in slower running engines, and is greatest at a point 
between the middle and the crank end of the rod. For this 
reason it is sometimes thought desirable to make the greatest 
diameter of rod, not at the middle, but somewhat nearer the 
crank end, as shown in Fig. 5'7. 

For moderate piston speeds this point need hardly be con- 
sidered as it is amply provided for in the co-efficient of security, 
but for high speeds and heavy ends it should be given due con- 
sideration. In the high speed type of engines such as the 
Porter Allen, the greatest strength of rod will be found at the 
crank pin end. At the same time, as will be seen, the value of 
■m, for high speed locomotive engines, is usually made small. 

For marine engines, m is usually taken quite high, viz.: 30, 

40, 60 or even So, and the ratio — =^ proportionally smaller. 

In such engines the rod is generally made proportional to the 
cylinder diameter, being about 0.0S5 to 0.095 times the bore. It 
must be remembered that in marine engines the stresses due to 
flexure of the hull, and general lack of rigidity, demand a 
higher co-efficient of security than for stationary engines. 




Fig. 518. 

In Fig. 518 is shown a rod for a screw propeller engine. The 
body of this rod is truly cylindrical, and the ends are similar to 
that shown in Pig. 500. 

Let E = 94,600 lbs. L = 60''. Taking, as before. 



Example 2.- 
have 



, w ^ 20 we 



D , 

— ^ = 0.0346 



i 



L 



D = 0.0346 v/94,600 



fi 



= 4.67". 



\/94,6oo 

In a similar case, executed by Maudslay, the rod was made 6" 
in diameter, which corresponds to a value in = 54.7. The 
diameter S, of the bolts in this case was 3", and according to 
the rule given for Fig. 499, they should be sji^'. 

^183. 

Rods of Rectangular Section. 

If it is desired to make the body of the rod rectangular in 
cross section, it is first necessary to determine the diameter for 
circular section by the methods of the preceding section, and 
then determine the equivalent rectangular section. 



THE CONSTRUCTOR. 



117 



Let: 

■ h, be the larger side of the rectangle, 
b, be the shorter side, 

(5, the diameter of the equivalent circular section at the 
same point ; then for a given value of /;, we have : 

('65) 



h 



A=^liLi = o.84^/- 

and for a given value of b : 

h 3 51- / rf \' / "J >' 



(166) 



and for a given ratio . 



4=</^i-o-</i " 



67) 



from which we deduce the following table : 



h 


b 


h 


b 


h 


b 


s 


& 


& 


& 


b 


S 


I.O 


0.84 


1.6 


0.72 


1.0 


0.88 


I.I 


o.Si 


1-7 


0.70 


1.25 


0.83 


1.2 


0.79 


r.S 


0.69 


1.50 


0.79 


1.3 


0.77 


2.0 


0.67 


1-75 


0.76 


1.4 


0.75 


2.2 


0.55 


2.00 


0.74 


1-5 


0.73 


2.4 


0.63 


2.5 


0.74 



If it is desired to calculate the rectangular section directly, 
without reference to the equivalent circular section, we proceed 





Fig. 519. 

as before, using the least moment of inertia of the section 

J = y'j /; i', and thus obtain for wrought iron or steel : 

for any given value of b : 

p TJi 
h = 0.0000000425 in 



for any given value of // : 

b = 0.0002 ^?'^;« ,/• 

V h 

and for any given ratio of /;, to b : 

h = 0.0144 ^^' /(4y /^^"^ 

For the last formula we have, when : 



(168) 
(169) 

(170) 



= 1.5 16 1.7 1.8 1.9 2.0 2.1 2.2 



2.4 2.5 



f(4J= 



1.36 1.42 1.49 1.55 1.62 1.6S 1.74 1.80 1. 87 1.93 1.99 



The most important application of flat connecting rods is upon 
locomotive engines. In this case the co-eiEcient of security is 
taken very low, i. e., the rod is made as light as possible, in 
order that the " stored velocity " maybe kept small, and the 
"whip" action reduced. 

An examination of practical examples shows values of m, 
from 2 to 1.5, taken at the middle of the rod. At the cross 




Fig. 520. 



head end the depth is reduced to o.S, to 0.7 that at the middle, 
and the depth at the crank end is that due to the taper thus 
indicated. An example of such a rod is shown in Fig. 519. 

h 
Example 2. — Given in a locomotive P =^ 28,600 lbs. L = 72" — j- 



have, if ;« = 1.5, according to § 1S2, ■%/ 

V^ 72 ■v/28,600 = 3.5" and 5 = 3.5 X 0-4 = i-lo say i V 



= 2.5. We 
i.i, hence h = 0.0144 X i.i X 1-99 
7 tt 



The " whip " action before referred to, is much more powerful 
in the parallel rods of locomotive engines than in the main 
connecting rods. Such a parallel rod, or side coupling rod is 
shown in Fig. 520. The keys for the boxes at each end of the 
rod are placed on the same side of the boxes, so that their 
action will not affect the distance between centres, providing 




Fig. 521. 

the wear is alike upon both ends, and for this reason it is 
desirable also that both pins should be of the same length. 
(See § 92.) In determining the cross section of such rods, it is 
to be assumed that the resistance offered is the same for both 
wheels. This mfeans, that for two coupled wheels, one-half 
the total driving force is exerted upon each ; for three wheels, 
two-thirds the total force is exerted upon the first rod, and one- 
third upon the second. At the same time it must not be for- 
gotten that under certain circumstances one of the wheels may 
slip. For this reason it is advisable to take a somewhat larger 
value for ;«, than for the driving rods. It is, therefore, not 
advisable to make ;«, less than 2, and if possible it should be 
greater, at least for two coupled wheels. If this is done there 
need be no fear that the rod will be excessively strained through 
slippage of wheels. 

Example 3. — The locomotive of the preceding example has two pairs ot 
coupled driving wheels. We have for the force transmitted through the 

coupling rod, P — ^^^ — 14,300 lbs. The length L'= 3 ft, 4 in. = 100", 

2 

and we will take the ratio — ,— = 2,5 as before. Taking fit = 2, we have 

from (170) h — 0.0144 X 1-19 X 1.09 "V 100 \/ 14,300 = 3.73" say 2%^'' This 
gives for b, 3.75 X 0,4 = i%". This corresponds closely with the proportions 
used on Eorsig's locomotives. Other examples in practice give values of 
m, as 1,9, 2. II, 2.8, etc. 

A rod of mixed section, passing from circular into rect- 
angular, is shown in Fig. 521, being the very elegant connecting 
rod of the Porter-Allen engine. In the illustration Z = 5 feet. 

?i84. 

Channeled and Ribbed Connecting Rods. 

Cast iron connecting rods are often made of cruciform or 
ribbed section, much in the same manner as axles. In such 

s 



B 
A 


F 


rrrrrrr 


J>i 


1 '1 


F 
B 








[ 1 








-::> 










C 

G 




-i.' 


41 


D 



cases it is best to determine an ideal round rod, according to 
Fig. 5, from which the desired section can be derived. 
For any given case, let : 

(5 = the diameter of the ideal rod, 

n, and b, the width, and thickness, respectively, 

then for any selected value of b, 



6 b '>\i6 ]( b\ h 

from which we get the following table ; 



(171) 



i 


b 


& 


b 


<s 


b 


6 


b 


& 


b 


"F 


h 


h 


h 


b 


h 


h 


h 


h 


h 


0.643 


O.IO 


0.700 


0.14 


0.748 


o.iS 


0.816 


0,25 


0.901 


0.36 


0-653 


0.1 1 


0.714 


0.1S 


0.758 


0.19 


0.S31 


0.27 


0.928 


0.40 


0.673 


0,12 


0.724 


0.16 


0.768 


20 


0.855 


0.30 


0.958 


0.45 


0.690 


0-13 


0736 


0.17 


0,789 


0.22 


0.872 


0.33 


0.987 


0.50 



Example i.— In Fig. 322, let A B CD, he the ideal round rod from which 
to construct a cruciform section ; E F G H, is the width selected for the ribs, 

theratioof?^ being, for example 1.5 We then have —— = —^ = o , 667, 
st h 1-5 

and this value in column i, oi the table gives for ___ , something between 

h 
II, and 12, This gives b = 0,12 h = o x2 S T. \i P Q = t.^pg, we have 
= 0.7 and in columns 3 and 4 we find b = 0,14 P Q. 



ii8 



THE CONSTRUCTOR. 



For constructive reasons the / section is preferred for loco- 
motive rods. Such a rod is shown in Fig. 523. This is made 
with a slight swell in the middle, but the scale of tlje drawing 
is too small to allow it to appear. 




Fig. 523. 

Such rods are either made with straight or rounded profile, 
as shown in Fig. 524. Neglecting the rounding we have for the 
least moment of iuertia of the section, 

/=Jj(2':^'+(// — 2f)i') 
For convenience of calculation we may, as in \ 163, assume a 

fir^ — ^ ''■"-" 

J, — B— -»J I, ...B- -J 

Fig. 524. 
rectangular section of a height //, and breadth b^, and then have 



from which, when the ratios . 



/; 



and are given, the nu- 

b 
merical values can be readily deduced. 

Example s. — A coupling' rod of /section, on a locomotive engine built by 
Krauss & Co., has the following dimensions: h = 3 149", b = 0.39", B ^ 1.85", 
c = o 6, L = 96.45, and /'= 10,890 lbs. To determine the degree of security 
/«, we substitute these values in (172) and obtain: 



</ 



°-6 / 1 V 



■ 1-325 



We then have from (i63) 



6 3A 



0.0000000 425 P Lr- 

_ (^-325)^ X 3-U9 



0.0000000425 X 10,890 y_ (96.5)2 
The completed rod weighed^only 125 pounds. 

2iS5. 

Forms of Cast and Wrought Iron Rods. 

In Figures 525 and 526, are shown comparative forms for a 
round connecting rod of wrought iron, and a cast iron rod of 
cruciform section. In the case of the cast iron rod, the fluted 





Fig. 527. 

portion terminates in collars near each end, the lower part at 
the crank end being made of flat rectangular section, enough 
longer than the crank arm to insure the necessary clearance. 

In Fig. 527 are shown some special forms for forked ends. 
Fig. 527 a, is a very short fork, Fig. 527 b, is for a flat wrought 
iron rod, and Fig. 527 c, is suitable for a long rod of cast iron. 
The boxes on these rods may be well secured by strap and key 
as in Sharp's pattern, Fig. 49S. In some cases connecting rods 
are made in the form of trussed frames, and the form of the 
ends are governed by the form of cross head used. The latter 
will be considered in the following chapter. 



ar 



Fig. 525. 




Fig. 526. 



CHAPTER XV. 

a?OSS H£ADS. 

? IS6. 

Various Kinds of Cross Heads. 

A cross head is that portion of a machine which makes the 

connection between the vibrating rod and the piston rod or other 

piece having a rectilinear motion. Cross heads are made with 

various kinds of journals, either overhung, forked or double; 

a b 




Fig. 52S. 



and in this respect are similar to the ends of levers, the differ- 
ence being that the path is curved in the one case and straight 



THE CONSTRUCTOR. 



119 



in the other. The path of a cross head is generally determined 
either by some form of parallel motion, or by guides, or in some 
cases only by the piston or other rod to which it may be at- 
tached. This gives the following classiiication : 

1. Free Cross Heads, 

2. Cross Heads for Link Guides, 

3. Cross Heads for Sliding Guides ; 

and this classification will be observed in the following discus- 
sion. 

Free Cross Heads. 

In Fig. 52S, a and b, -are shown two forms stiitable for small 
free cross heads. These are made with double journals of 
wrought iron. The diameter of the piston rod in the cross 




Fig. 529. 

head should not be less than d,. A modification of this form is 
shown in Fig. 529. Satisfactory proportions will be obtained by 
making the height li in the middle equal to 

k = 2.sd,-\-r-- (173) 

4- 

in which A is the length of arm ; also for the thickness b, which 
is uniform, 

PA 
b = 0.00035 —jjT (174) 

The curve of the profile may be made as shown in ^ 142. 

ExavipU 1. Given the load P = SSoo lbs., and the length of arm A = 15.75" 
for a cross head, as in Fig. 529. According to the table of § 90, we have d<^ = 
0.027 \/ P = 0.027 \/ 4400 = 1.35". 

We have from (173) 



h = 2.5 X 1.85" + ■ 



and from {174) 



14 



= 5-75 ' 



b = 0.0 



SSno X 15.75 
' (5.75)= 



= 1-47", say ij<". 



The other dimensions as given in the figure are; Hub thickness 0.5 t/o = 
0.5 X 1.S5'' = 0.925'', say i" ; depth of key = 0.67 X 1.85 = i J^" ; thickness" ot 
key = 0.2 X 1-87" = %". 

Example 2. The engine of the steamship " I.a Plata " has steam cylinder 
103" diameter, with a maximum steam pressure of 26 pounds per square 
inch, giving a total pressure of about 217,000 pounds on the piston rod. The 
length A is 68", and in the executed engine the builder, Napier, has made 
h = 28", b = 7", (/o ^ ^o". the length of journal = 15", these latter agreeing 
closely with those obtained from ^01. The hub length was made — 30", and 
hub thickness 5", with a bore of 10". According to the above formula, 
we get (fa = 8.75", A = 27", * = 7%"- 




Fig. 530- 

I iSS. 
Cross Heads for Link Connections. 

Cross heads which are intended to be guided by a sj-stem of 
linkages or parallel motions are made with a pair of link jour- 
nals in addition to the journals for the connecting rod, and the 
former are generally made as prolongations of the latter. 

In Fig. 530 is shown a wrought iron cross head for use upon 
a beam engine in connection with a Watt parallel motion. The 



unit upon which the dimensions are based is 

(/j ^ 0.026 \/ /^ 4- 0-2 (17s) 

in which Pts the total load on the cross head. The same mod- 
ulus serves for the simple proportions of the following cross 
heads. The load F^ upon the link journals can be determined 
from the load P^ of the rod journals by the following relations : 

P, sin a 



A 



' cos /i 



(176) 



in which a is the greatest angle which the connecting rod makes 
with the axis of the piston rod, and jS the angle which the link 



.3/.5a2.-^„ 




Fig- 531- 

makes with a normal to the axis of the piston rod when « is a 
maximum, the latter position being determined most readily 
from the drawing. 

ExamfAe. If the aug'Ie a at its Tiiaximuni is 20°, and the corresponding' 
value of ^ =^ 15°. we have 

sin a 0.3420 

" 0.9659 ~ °'^^' 



cos j3 



hence /^ = 0.35 P>^. 

When the connecting rod acts directly upon a crank the angle a is usually 
20° or more, but when the connection is to a beam it is seldom greater than 10°. 

Another form of wrought iron cross head for link connections is shown in 
Fig. 531. Thisfoim is especially convenient when occasion requires that 
the piston rod be disconnected readily, and is especially adapted for direct- 
acting steam engines. 

? 189. 

Cross Heads for Goides. 

Cross heads for use with guide bars are made in many varied 
forms for steam engines and pumps. The form is modified to a 
great estent by the number and arrangement of the guide bars. 

Fig. 532 shows a much used form of cross head for four guide 
bars. If the engine runs constantly in the same direction, and 
the pressure upon the piston acts alwa}-s in the direction of its 




■t-^ 



Fig. 53 



motion or in the opposite directiou, the pressure will be almost 
entirely confiued to one pair of guide surfaces, the other pair 
only coming into action in the case of extraneous forces. If 
the pressure acts sometimes with the direction of motion and 
sometimes against it, the result will be to cause the pressure on 
the slides to alternate. In most steam engines the pressure 
changes not only in directiou but in magnitude, especially near 
the end of the stroke. The slides should be made of a softer 
material than the guide bars in ordei that the greater wear may- 
come upon those parts which are most easily replaced. In order 
to reduce wear it is also desirable that the surface of each slide 
should not be less than 2.5 P\ /^ being the total pressure on the 
piston in kilogrammes, and the area thus obtained being in 
square millimetres. This is about equivalent to 0.0018 P\ P be- 
ing the total pressure in pounds, and the area given in square 
inches. Many use double this area, or 0.0036 P, with corre- 
sponding reduced wear on the parts. The pressure on the sur- 



I20 



THE CONSTRUCTOR. 



face of the slides, witii the ordinar)' ratio of connectiug rod to 
crank arrc, will then be about i3o pounds per square inch in the 
first case and about 60 pounds in the second. 

If we represent the superficial pressure, rubbing Velocity and 
coefficient of friction for slide and crank pin respectivelj' by 
At A. ^'i' ^'2>/i-fi< '^^ have for the lineal wear per second : U^ '= 
fiPi ''■'\f\ ^'^'i ^i ^^ I'-iPi ^2/2. ill which y-i and ji^ are coefficients 
due to the materials used. Some of these values vary at differ- 




FiG. 533- 

ent portions of the [stroke. If, however, we take them at the 
same instant, we have the ratio of wear for that point, 

U\^ ^ I'lPiA'^'i 
U-i HPifi ^2 
The point of maximum wear upon guides is near the middle 

2 Tt R n -K d n 

of the stroke, where v, = and zu = 7 

1 60 X 12 60 X 12 

Taking the values of /^ and U the same in both cases, we ob- 
tain, by substitution in the preceding equation, 

A d_ 

p". ^ R 

which gives an average ratio of about yV, and taking /j at 1420 
pounds gives about 120 pounds for/,. If we consider the pres- 
sure on the pin to be alternating and that on the slides contin- 



— ,. 


" -^\ 


a. 


^ 1 


' 


1 


? 


[ 




:f i'' 


-o>--~- 








—~ 




1 




V/ - 




" 


1 ■' 






Fig. 534. 

nous, p., becomes onlj' 710, making />, about 60 pounds. If the 
ratio of connecting rod to crank arm is unusually small, the 
pressure Q on the slides at mid-stroke should be calculated, and 

It may be taken as C? = j 

The cross head shown in Fig. 533 is arranged for a fork jour- 
nal, the latter being also in this case made spherical. The fork, 
which is keyed to the piston rod, is intended to be made of 
wrought iron ; should it be made, instead, of cast iron, the 
thickness of the metal about the hub should be increased to 
o.2Srfi, and its length to \.T$d^. This form permits the slides 
to be brought closer together than in the preceding design. 

A very simple form of cross head for four-bar guides is used 
on man}' American locomotives, as shown at a and b^ Fig. 534. 
For constructive reasons, to obtain the necessary clearance, this 
form is sometimes made as at b, with the middle plane of the 



guides above the axis of the piston rod. The cross head is of 
cast iron, with the pin cast in, and finished by special machin- 
erj'. A similar form of cross head to that shown at a is used 
on the Porter-Allen engine, except that a steel pin is inserted as 
shown at c. The flattening of the top and bottom of the pin 
serves to assist in the distribution of the lubricant. "■■ 

The area of slides in America is about that given by the fore- 
going rule. 

Example. A wood-burninjj passenger engine has cylinder 16" diameter, 
at no pounds pressure, giving P= 22,110 lbs. The surface of each slide 
measured 79 square inches, or about 22,110 X 0.0036 = 79. 59 sq. in. 

The forms of cross head shown are generally fitted with slides 
of white metal or bronze, and in some instances bearing surfaces 
of glass have given good results. 

There is one form of marine engine which requires a special 
form of cross head. This is the so-called back-acting engine, 
in which the crank shaft is placed between the cjdinder and the 
cross head, and there are two piston rods, passing above and 




Fig. 535. 

below the shaft. There have been man}- varieties of this tj'pe 
constructed. In Fig. 535 is shown a design by Maudslay. The 
body of the cross head is formed like an axle, with two project- 
ing bosses for the attachment of the piston rods. The distance 
E is governed by the diameter of the crank shaft, and A by the 
clearance space required for the crank arms. In this design the 
slides are placed outside of the piston rods ; other builders, as 
Ravenhill, place them between the rods and the journal d' , 
where, as will be seen, there is sufficient room. The lower por- 
tion of the slides are made of bronze and fitted with adjusting 
keys. The dimensions of the body are obtained by considering 
it as an axle, remembering that the forces act to produce twist- 
ing with the arm E as well as bending with the arm A. The 
length I,' is to be taken in connection with the diameter d' , so 
as to keep the pressure on the journal within practical limits. 
English practice in such construction gives pressures ranging 
from Soo to 1800 pounds per square inch. The diameter 6 of the 
threaded ends of the rods is the same as given for Fig. 499. 




Fig. 536. 

In Fig. 536 is shown Stephenson's cross head. Here the 
guides are brought so close together that each pair merge into 
one, and there are but two guide bars. The middle piece, ot 
wrought iron, is made with two journals, for a forked connect- 
ing rod. The slides are best made of bronze, the area being as 
before = o.oojb P, except in the case of locomotives, where the 
limited space often causes it to be reduced to o.ooiS /". 

Another design for double guide bars is that of Borsig, shown 
in Fig. 537. This contains a fork journal, whose projected area 
/' X d' should not be made too small. Sometimes this is made 
so small that the pressure reaches 3000 to 4000 pounds, and hot 
bearings and cut boxes are apt to lollow. Judgment in this re- 
spect is most important for all bearings. The slides are made 
of cast iron, with bronze shoes, which are packed out with thin 
slips of copper or zinc. 



* This is also done on a horizontal engine built by Brown, of Wintherthur 
See Engineering, Jan., 1880, p. 70. 



THE CONSTRUCTOR. 



121 



Fig- 53S shows a noteworthy form of cross head used ou thft 
Western Railway of France. The body is of wrought iron, the 
slides and piston rod connection are of steel. The manner in 




Fig. 537- 

which the reverse taper of the rod is secured, by means of a key 
and conical ^hell of steel, is of peculiar interest. Since this is 
a special construction a few dimensions are given in the figure 
(in millimetres). The area of the slides does not appear to be 

^ ^ 




large. The rather complicated form of the head of the pin is 
shown in the lower right hand corner of the illustration. 

Fig. 539 shows a cross head of the so-called "slipper " tj'pe, 
for single guide bar. This is well adapted for situations in 
which the direction of rotation is constant and the pressure al- 
ways downward. In order to provide for possible lifting forces, 
and to meet the reverse action of compression and inertia, the 
beveled shoes are used, although a square shoulder is to be pre- 
ferred. The area of slide should not be less than 0.0036 P, pref- 
erably more.'- 

Another form of cross head for single guide is given in Fig. 
540. This is from a marine engine by Humphrey's, Tennant & 
Co., and is intended to serve for pressure in either direction. In 
this case the bearing is in the cross head, and the pin is intended 



>. 0'<5a>-t US: 




Fig. sag- 
to be fast in the connecting rod, being attached as in Fig. 537. 
The wear on the bearing is taken up by the removal of thin 
slips of copper, originally placed in the vertical joint ; and wear 
upon the guide, by the insertion of similar slips between the 
cross head and slide. The whole construction is applicable to 
many situations. The middle portion is in this case made of 
bronze, but may be of cast iron, when the bearing is lined with 



white metal. The modulus for the dimensions is the same as in 
formula (160), and the bolt diameter rf, as in Fig. 499, using for 
d the diameter of the equivalent normal wrought iron overhung 
pin. 

A somewhat similar cross head has been designed by Napier 
for use with the horizontal back-acting marine engine, Fig. 541. 
This is intended to be used with a forked connecting rod. The 




Fig. 540. 



middle block is made of cast iron, and the distance B is kept as 
small as possible, in order to reduce the size and weight. The 
depth of arm h is determined as in Case I or II, ^ 6. The bolts, 
whose diameter 6 is calculated as for Fig. 499, are secured by 
jam nuts. 

In Fig. 542 is another excellent design by Maudslay for simi- 
lar service. This is for an ordinary connecting rod, as in Fig. 
5 1 8. The pin is formed in the crooked wrought iron piece which 
also forms the arms. The thickness h' of the latter is deter- 
mined from the corresponding moment after having selected 
the depth /;, which in this case is made equal to d' . The value 




Fig. 541. 



of d' is calculated as in the case of an axle. The screw diam- 
eter i' is calculated as before, and should be made full}- as large 
as the formula gives. The small lug on the lower part of the 
right arm is for the attachment of the pump rod. Such attach- 
ments are frequently made to the cross heads of marine engines, 
of which this is a good example. On the left the slide is shown 
in section. This is cast of bronze, with the channels shown 




* For a similar cross head, designed by Stroudley, for locomotive sen-ice, 
see Engineering , Feb., 1867, p. 65. 



Fig. 542. 

filled with white metal. The small shoe on the right, which is 
secured by screws, can be removed, so that slips of thin copper 
can be inserted to take up for wear. These last two cross heads, 
although unusual in appearance, show how a difficult construc- 
tive problem can be solved completely, and may be regarded as 
types. 

? 190. 

Guides and Guide Bars. 

Guides are made of wrought iron, cast iron or steel. If the 
entire pressure comes upon one guide, as in the designs just de- 
scribed, and the guide is supported onlj' at the ends, which are 



122 



THE CONSTRUCTOR. 



separated by a distance = jj + i,, it must be calculated to resist 
bending. Taking the crank at right angles to the guide, as the 
most unfavorable position, and calling the pressure Q, and the 
distances of the two points of support from the centre of the 



i" 






TlflB iillllllliilliilMM Ii^^ 



Fig. 543. 

cross head as s-^ and s.^, Fig. 543, we have the bending moment 
of the bar = Q — ~= — , and for the relation between the depth 



and width of bar : 



Jl + s., 



■'= /I 



■'1 + -^2 



(177) 



Q 
S 6 s^ 

The permissible value of stress 6" for wrought iron or steel 
should be small, say 7000 pounds, in order that but little deflec- 
tion shall occur. Any springing is especially hurtful in this 
case, since it prevents the entire surface of the slides from bear- 
ing fairly, and thus causes greatly increased pressure upon the 




Fig. 544. 

points vfhich are in contact. Deflections of ^V" or more are 
sometimes found, w^ith corresponding irregular wear upon the 
slides. This subject can be tboroughlj' investigated graphically 
by taking the various positions of the load. 

In Fig. 544 is shown a form of cast iron guides, intended to 
receive pressure only upon the lower guide. This is only sub- 
ject to compression, and hence very little deflection can occur. 




Fig. 545. 



The sectional view on the left shows the disposition of the ma- 
terial, and it will be noticed that the flanges on the cross head 
are arranged so as to retain the oil. The upper guide is bolted 
to the lower, and should the motion be reversed, throwiug the 
pressure on the upper guide, the bolts must be made pro|3ortion- 
ally stronger. 

A form of guides which is coming more and more into use for 
stationary engines is that shown in Fig. 54.5. Here the flat 




Fig. S46, 

guide surfaces are replaced by portions of a cylinder. An espe- 
cial advantage of this construction lies in the possibility of bor- 
ing the guide surfaces in exact alignment with the cylinder. 
Any twisting of the cross head is prevented by the connecting 
rod and crank pin, or, if necessary, a tongue on the lower slide 
may fit into a groove in the guide. 



The cross head for such guides may be similar to Fig. 537, the 
lower guide being adjusted b}' a key. 

The single guide bar has been used in locomotive practice. 
Fig. 546, which was shown both on American and Belgian en- 
gines at the Paris Exposition of 1S78. The guide is bolted to 
the cylinder at C, and to the yoke at _/. The cross head is a. 
simple modification of the form in Fig. 5346. Engineer J.J. 
Birckel has shown that there is a heavy lateral stress on such a 
guide bar, due to the necessar}' end plaj' in the driving axles, 
and a wide bar is therefore necessary. He makes the width. 
f> = 2-< /;, and makes 



// = Const 



7 






in which G is the weight of the parts subject to lateral vibra- 
tion, O the normal component of the piston pressure, X the 
length of guide bar, and // the distance from centre of bar to 
centre of rod. In the case of a cylinder iS" diameter at 100 
lbs. steam pressure, G = SSoo lbs., /, = 51.2 and // = T-S''^ 
the values obtained are : b =^ S'^, h = 3". 




Fig. 547- 

Fig. 547 is a cast iron guide for horizontal marine engine, 
suitable for a cross head such as is shown in Fig. 540. This is 
especially arranged to retain the lubricating oil, and as the 
cross head moves between the positions i' — i and 2 — 2', 
every stroke, it dips in the oil at each end and carries it over 
the guide. 

Example. The steamship " Arizona " is fitted with single guide bars and 
automatic lubrication. The pressure on one slide is 64,000 lbs., the area 
being 47" X 27" = 1269 sq. in., or a pressure of about 50 pounds per inch. 



CHAPTER XVI. 

FRICTION WHEELS. 

I 191. 

Classification of Wheels. 
Wheels are used in many varied ways to transmit motion in 
machine construction. They may be divided into two great 
classes : 

1. Friction wheels, 

2. Gear wheels, 

according as they transmit motion by frictional contact, or by 
the engagement of gear teeth. 
Each of these classes may again be divided into : 
(a) Direct acting, and 
(5) Indirect acting wheels, 
according as the force is transmitted directly from one wheel to 
another, or indirectly, by means of belt, cord, chain, or similar 
device. This gives four divisions for consideration, as follows : 
I. Direct Acting Friction Wheels, or friction gearing, pure 

and '"•imple. 
II. Direct Acting Tooth Gearing, otherwise called simply, 
gearing. 

III. Indirect Acting Friction Wheels, such as Pulleys, Cora 

Wheels, &c. 

IV. Indirect Acting Tooth Gearing, such as Chain Wheels. 
The first three forms exhibit the greatest variety, and will be 

given the first consideration. 

The relative position of the axes has a most important influ- 
ence upon the form of a pair of wheels. The positions may be 
grouped as follows : 

1. The axes geometrically coincide, 

2. They are parallel, 

3. They intersect, at an angle, 

4. They are at an angle, by pass without intersecting. 

This gives four groups under each of the preceding main divi- 
sions. 



THE CONSTRUCTOR. 



i 192- 
Thb Two Applications of Friction Wheei<s. 

Direct acting friction wheels may be used to accomplish either 
one of two different functions and their construction varies ac- 
cording to the use to which they are put. 

The first application is that in which the wheels are pressed 
together with sufficient force to prevent the surfaces from slip- 
ping upon each other, under which circtimstances the motion 
of one wheel will be transmitted to the other. 

The second application is that in which the so-called rolling 
friction is so small that the wheels, when interposed between 
two surfaces which are relatively in motion, act to reduce the 
otherwise injurious frictional resistance. 

Hence we see that friction wheels may be used : 

(a) To transmit motion, and 

(b) To reduce resistance. 

The first application includes whal> may be called driving 
friction wheels, or commonly simple friction wheels, and the 
second application includes all the various forms of friction 
rollers, roller bearings, ball bearings, and the like. The two 
kinds have also been termed friction wheels and anti-fi-iction 
wheels. 

5 I93' 

Friction Wheels for Parallel Axes. 

The surfaces of a pair of friction wheels in contact are almost 
always of circular curvature, and when a pair of such wheels 
roll freely upon each other the number of revolutions will bear 
an inverse relation to the radii of the respective circles. This 




Fig. 54S. 



ratio is called the velocity ratio of the wheels. If we call the 
revolutions per minute of each wheel n for the driver and n^ 
for the driven wheel ; and the corresponding radii R and R^, 
we have for the velocity ratio : 

?=| (-»> 

Friction wheels for parallel axes are made with cylindrical 
surfaces. Fig. 54S. In order that there shall be no slipping be- 
tween the surfaces we must have a pressure O, which, to transmit 
a force P, at the periphery of the wheels, must not be less than 

Q = J ('79) 

f being the co-efficient of friction. 

Ihe value of_/for various materials may be taken as follows : 

For Iron on Iron o.io to 0.30 

" Wood on Iron o.io to o.5o 

" Wood on Wood 0.40 to 0.60 

Friction driving is often very simple and practically effective 
It had been almost neglected for general uses, when it was very 
successfully applied in various forms of saw mill machinery. 
This was especially the case in the lumber regions of America.* 

The best results are obtained in practice from surfaces of 
wood on iron, the wooden surface being preferably the driver, 
so that any stoppage on starting shall not wear hollows in the 
softer material.! The rim is built up in such a manner as to 
place the grain of the wood as nearly as possible in the direc- 
tion of the circumference. The best wood for the purpose is 
maple, but linden, poplar and pine have been used with good 
results. Great care must be taken to make the wheels truly 
cylindrical, and they should be keyed upon their axles and fin- 
ished while running in their own proper bearings. Under these 
conditions a wheel of maple can transmit a circumferential force 

* See Wicklin, "Frictional Gearing," Sci. Am., vol. 26, p. 227; also Apple- 
ton's " Cyclopedia of Mechanics," vol. 2, p. 36 ; also Cooper's " Use of Belt- 
ing," p. 288. . 

t Surfaces of compressed paper against iron are now m general use.— 
Trans 



of about 2S pounds per inch of face width, or from 15 to 20 
pounds for the other woods above mentioned. 
This gives for maple face : 



■ 28" 



iiSo(///^) 



(180) 



and a width i >< to 2 times greater for the other woods, II P 
beingthe horsepower transmitted, and v the circumferential 
velocity in feet per minute. Substituting for v its equivalent 



value, 



■ Rn 



we have 



2414 HP 



R 



(.81) 



Such wheels are made in practice up to 6 feet in din meter and 
30 inches face, transmitting upwards of 60 horse power. 

According to the experiments of Wicklin, the coefficient of 
friction is about 0.30 to 0.32, from which the pressure of contact 
must be (9 = 3 '-3 P. The ease with which these wheels can be 
thrown out of gear is a very convenient feature. 

Example I. Let 10 H. P. be required to be transmitted by friction wheels, 
the speed of shaft being 80 revolutions per minute, and a circumferential 

velocity of iiSo feet per minute given. AVe get from (180) b = — ^ . 10 = 10" 



face, and from (iSi) R = 



2414 X 10 



If the driven shaft is run 100 rev- 



10 X 80 
olutions per minute, the radius of its wheel will be R\ = 30'' X o-8 = 24". 

Example 2. Required to transmit i H. P., the given value being n = 90^ 
«i = 75, y? = 12", R = 13.66". From (181) we have 
2414 



*= . 



: X90 



= 21/". 



If pine is used, this should be doubled, giving b = 4j^". 

The method of construction of these wheels is as follows : 
For large wheels, 4 to 10 feet in diameter, the rims are made 
from 6 to 7 inches deep, built up of wooden segments ij/in. 
to 2 in. thick, forming "a to yj the circumference, and so placed 
that the direction of the fibre shall follow the circumference of 




!< 




Fig. 549- 



Fig. 550. 



the wheel as nearly as possible. These segments are firmly 
clamped together and secured by bolts or nails. The actual 
face is made about 2 in. narrower than the working face b. This 
rim is then securely fastened to the arms, which are ver}' strong 
and made with feet or pads which are mortised into the rim and 
both keyed and bolted fast. The number of arms varies from 6 
to S, and for very wide faces two sets are used ; see Fig. 549. 
An additional ring of wood is then put on each side, bringing 
the width up to the full value of b, and these outer segments 
are deeper than the others, so that the ends of the keys are en- 
tirely covered ; the completed wheel is then turned and finished 
in place, as before stated. 

Smaller wheels are built upon iron drums, the segments being 
screwed together and clamped between the outer rims. Fig. 550. 
Projections on the iron rim, let into wood, prevent the latter 
from turning. The total thickness of rim is about 4 in. Care 
must be taken that the wood is thoroughly dry. 

The driven wheel of iron is made similar to a belt pulley, but 
with a much stronger rim and more and heavier arms ; when a 
wider face than 16 in. to iS in., double arms are used. Both 
wooden and iron wheels should be carefully balanced, in order 
to avoid vibration. , 

An important and ingenious use of friction wheels is in con- 
nection with a drop hammer, the wheels being used to raise the 
drop. Merrill's drop hammer, Fig. 551, is operated by two iron 
friction wheels A and C, which together act upon the oak 
plank B, to which the hammer drop is attached. The roller A 
is the driven one, and its shaft runs in eccentric bearings on 
each side, which are operated by levers D and press the parts to- 



124 



THE CONSTRUCTOR. 



gether. When the parts are in the position shown, the plank 
and hammer are raised, and when the lever D is lifted, the 
wheels separate and the hammer is allowed to drop. In some 

I 




Fig. 551. 

similar designs both rollers are driven, as in the hammer of 
Hotchkiss and Stiles,* and also in the so-called " Precision 
Hammer," of Hasse o: Co., of Berlin. f 

I 194. 

Friction Wheels for Inclined Axes. 

When the axes are inclined to each other, the surfaces of the 
•wheels, unless they are very narrow, become portions of cones, 
•with a common apex at the intersection of the axes. Fig. 552. 
Each pair of circles in the surfaces then roll together as if cyl- 
indrical. Wheels of this sort may be constructed in a similar 




Fig. 552. 

•manner to those described in the precediug section. In Fig. 
553 are shown, at a and b, t-n'O sizes of conical wooden friction 
wheels. The outer disk is placed with the fibres in a radial' 
direction, but the others have the grain of the wood arranged as 
nearly as possible circumferentially. These disks should be 
most carefully fitted, glued and bolted together. Especially im- 
portant is it that conical surfaces should be turned to the cor- 




FiG. 553- 

rect angle. The pressure is applied from the end of one of the 
two shafts in such a manner that the force may be applied or 
removed at the thrust bearing. 

The most extensive application of friction driving, both with 
cylindrical and conical su'cfaces, is found in locomotive engines. 
The high pressures necessarily used compel in this case the use 
of iron or steel tires. The force Q here exceeds 6 tons.j 

In some cases a combination of one conical wheel and one 
narrow wheel with rounded edge, as in Fig. 554, may be used for 
the transmission of small po-s\'ers. In this case both wheels are 
made of iron. The pressure is easily applied to the disk wheel 
B, and the mechanism is so arranged that it can be shifted 
along its axis, so that a variable speed motion is obtained. It 
must be noted that in this form the surfaces in contact are ne- 
cessarily very limited, and hence it is desirable, as in the case of 
friction couplings, to have the diameters as large as possible. 



* See .\ppleton's " Cyclopedia of Mechanics," vol. 2, p. 85. 

\ German Patent 2685. In this hammer the lower part of the plank is re- 
duced, and the whole design very ingeniously worked out. 

\ The surfaces in contact are sensibly flattened. Krauss' experiments 
showed that with a pressure of 12 000 pounds, a steel tire on an iron rail gave 
a surface of contact of 0.309 sq. in., and with a pressure of S250 pounds, a 
surface of 0.24 sq. in. In the Foutaine locomotive the pressure of contact 
■was about 8 tous on each wheel. 



and the linear velocity high, in order that the driving force may 
be kept as small as practicable. The most convenient modifi- 
cation of this form is that in which the angle B of the cone is 
made 180°, when we obtain a pair of friction disks. Fig. 555. 

The velocity ratio, when A is the driver and B the driven, 
and X is the distance from the axis of «, is expressed by : 



i=^^i^, which = 



r 



(1S2) 



when /3 ^ 180°. The change of velocity is expressed hy the 
line O N. \i B\s the driver and A driven, we have 



X sin /3 



, which 



n 



(183) 



when ,9 = 180 ; n being the number of revolutions of B. These 
are the equations of an equilateral hyperbola; see Fig. 555. 
When the value oi x approaches near zero, the driving oi A by 
B becomes impracticable.* 




Fig. 554. 

In Fig. 556 is shown a form of variable speed gear in 
which one disk is placed between two others. The disks A^ 
and A.,_ revolve with the same velocity in opposite directions, 
and the driven disk B is placed between. The velocity ratio 

can be varied from o to ^- proportional to .-tr.f The pressure 

is applied at the ends of both horizontal shafts. This arrange- 




-" ^■— 



i'Ea] 



Al 



-j- 

j- 



Fig. 555. 



Fig. 556. 



ment has been used for driving centrifugal machines, and more 
recently for potters' -n'heels, the control over the speed being 
especially useful in the latter case, the position of the variable 
disk being controlled by a treadle. 

Another arrangement of disk friction wheels to produce a 
variable speed is that of Rupp, shown in 557. A is the driver, 
B the driven, and C the intermediate, the latter being ad- 
justable on its axis. The variation is bet-sveen the limits 

a—R -, R 

and 

R a—R 

according to the relation 

?/[ X 

11 a — X 



{1S4) 

which gives the equilateral hyperbola shown in Fig. 557, inter- 
secting the axis of ordinates when jr ^ o. Rupp recommends 
especially that the intermediate wheel be made of a number of 



* In the variable speed gear of Lecoeur (Gernian Patent 17,078) a loose disk 
is filled ia the centre of --/, so that if B approaches too near the centre the 
motion ceases. 

t See Berliner Verhandlung, 1S66, p, 39. This arrangement has been used 
especially for regulating" the speed of cotton-spinning machinery. 



THE CONSTRUCTOR. 



125 



thin disks, all loose upon the shaft. This does not appear to be 
advantageous in view of formula (1S4), since there is a different 
ratio for each disk, and hence some of them must slip. 

A similar device is that of Barnhurst, Fig. 558, in which the 
disk is placed between two cones,* 




By making two of the disks fast on one shaft, and placing 
the driving wheel between them, with sufScient clearance to 
enable either to be brought in contact with the driver, the driven 
shaft may be operated in either direction or allowed to remain 




Fig. 558. 

at rest. Fig. 559. A-y A^ are the driven, and B the driver. This 
is ingeniously applied in Cheret's Press, in which the screw of 
the press is on the axis of B, and is turned in either direction 
by the friction wheels. 




Fig- 559- 

I 195. 

Friction Wheels with Inclined Axes not Intersecting. 

In the case of friction wheels whose axes are rigidly held, and, 
while inclined, do not intersect each other, there is always more 
or less lateral slipping. The figures which, under these condi- 
tions, exert a maximum amount of rolling action and a mini- 
mum of slipping are a pair of h3'perboloids of revolution (see 
?2i8). If, however, the axes are so arranged as to permit 
longitudinal motion, either with the bearings or in them, the 
■wheels will be relieved from slipping. Such an arrangement, by 
Robertson, is shown in Fig. 55o.-(- The disk A acts upon a cyl- 



inder B, the axis of which makes a small angle with that of A. 
When the disk A is revolved, it rolls a helical path upon the 
cylinder, and also moves in the direction of its axis. The angle 
a corresponds to the angle of the screw thread. Robertson has 
applied this device as a feed motion to a wood lathe. This ar- 



TZl" 




Fig. 560. 

rangement ma}' also be reversed, A being held in its bearings, 
and B, with its bearings, permitted to travel. 

The same principle may be used with cones on disks, but 
these devices appear to possess limited practical application. 

Friction wheels, the axes of which coincide, are the same as 
friction couplings. 

? 196. 

Wedge Friction Wheels. 

Wedge friction wheels are those in which the cross section of 
the rim is wedge-shaped. They were designed in Italy by Mi- 
notto and in England by Robertson, and hence are known by 
both names ; in both cases being applied to wheels with parallel 
axes. Two forms of rim section are given in Fig, 561. In this 
case the radial pressure Q is much less than with cylindrical 
wheels, and for any wedge angle Q it is equal to 



Q=P 



sin \- f cos 

7 







(185) 



A disadvantage of this form is the fact that true rolling action 
only takes place in one cylindrical section through each rim, 
and hence there is much hurtful friction from the slippage at 
other points ; this defect becomes less as the ratio of the wedge 
depths k, k-^ to the radii Ji, J?^ diminishes.- In order that the 

k k^ 

ratio -77 and ^ may be kept as small as possible without re- 
ducing the surface of contact, the rim is made with multiple 
grooves, as in the form on the right. The angle 8 is generally 
made ^ 30° although Robertson used much smaller angles. 



krJi-l], 




Fig. 561. 

These wheels grow warm and wear rapidly when operated con- 
tinuously at high speeds. Minotto has also made especial ef- 
forts to design bevel wedge friction wheels ; he uses only one 
groove, and adjusts the position so that wedge profile shall al- 
ways act at the same point. Robertson makes the grooves non- 
adjustable, as in spur wheels. Wedge friction driving has been 
proposed for locomotive driving, and models made on this plan 
have ascended steep grades ; the wear in this case comes mainly 
upon the track. 

Wedge friction wheels have been used in America for many 
years on winding engines ; and they are especially useful in 
driving ship's windlasses, on account of the ease with which 
the}' can be thrown in and out of gear.t More recently wedge 
friction wheels have been used by Gwyune and also by Weber 
in Berlin, at high speeds, and apparently with good endurance, 



* See Engineer^ June, 18S0, p. 404 ; also H. Kdnig, German Patent No. 9365. 
t See En^neer, 1867, p. 410, in which many interesting designs by Robert- 
son are given. 



* Hansen, in Dingier^ s Journal ^ vol. 137, 1855, p. 1, shows that the actual 
rolling circle is always on that portion of the wedge surface towards the 
driving-wheel, and chano;es its position when the driver becomes the driven. 
See also Ad. Ernst, in Zeitschr. d. V. deutscher Ingenieure, xxvi, p, 243. 

t H. D. Andrews' steam windlasses are made with wedge gear of from 4 to 
12 grooves. The diameters of the friction wheels are as follows; 

H. P. Slow speed. Fast speed. Drum, 

Diam, Length, 

5 4 — 30" 8 — 26" 6" 27" 

S 4—30" 8—26" 8" 27" 

10 6—36" 12—30" 8" 30" 

15 6 — 36" 12 — 30" 8" 30" 



126 



THE CONSTRUCTOR. 



driving centrifugal pumps at 700 revolutions per minute. These 
wheels are with single groove and wedge, the wedge being of 
curved profile, and hence acting somewhat like the adjustable 
device of Minotto.* 




Fig. 562. 

Single-groove friction wheels have also been used in America 
for mill gearing. 

Sellers has devised an ingenious form of wedge friction gear 
for changing the rate of feed on engine lathes. This is com- 
posed. Fig. 562, of two simple disks and a pair of ver}' obtuse 
cone plates, the latter being pressed together by springs. The 
axis of the cone plates is movable, thus giving change of speeds. 
The ratio of change is similar to Rupp's gearing, formula (184). 

§'97- 
Speciai, Applications of Friction Wheels. 

The previously stated condition of wedge friction wheels, that 
there is but one line at which rolling action takes place, and that 
slipping occurs at all other points of contact, is utilized in vari- 
ous methods in machine design, as for example, in rolling mill 
machinery. 

In this case a third piece is driven, compressed and altered in 
form between two friction rolling members. The rolls and the 
metal may be considered as a train of friction gearing. In the 
case of a plate mill, the plate may be considered as a pair of 
friction wheels of infinitely great radii ; this is also the case in 
rolling bars. In a tire mill one surface is an internal and one 
an external wheel, of variable radius. The three-high mill may 
be similarly compared to a train of friction gears. 




Fig. 563. 

A very interesting application is that referred to in ? 14S, as 
in use at the Kirkstall Forge, and sho%vn in Fig. 563. A and B 
are plane friction disks. The round bar C passes between them, 
slightly above the centre and partlj' rolling, partly sliding, re- 
ceives both an endlong motion and a motion of revolution upon 
its axis. The disks revolve in the same direction, and of the 
opposed forces which tend to cause revolution of the bar those 
which act in the portion of the disks between their axes, i. e., 
between the vertical dotted lines in the figure, preponderate, 
and determine the direction in which the round bar revolves. 
The horizontal components of the sliding forces at all portions 
of the disks, act to carry the bar forwaril, so that it receives a 
combined spiral motion and is at the same time rolled and 
straightened. The earlier method of rolling round bars was by 
means of semicircular grooves, but this does not give either as 
round or as straight a product. Many similar examples in roll- 
ing mill machinery will be found, resembling friction driving 

In the same way, various forms of grinding mills are made 
upon the principle of friction combinations, as in the case of the 
Eogardus mills, with flat grinding disks, and also in the case of 
grinding rollers, Fig. 564. Here the round trough A revolves, 



* See Engineerings iS68, pp. 502, 593, and 1869, p. 353. Engineer Brauer, 
assistant in the Royal Technical Pligli School, lias attempted to adapt the 
principle of the Weston Clutch (g 157) to friction wheels. The wheels are 
made of a number of thin plates, with rubber washers between them, and 
a slight axial pressure is sufficient to cause them to grasp each other with 
much friction. A description will be found in Berlin Verhandlung, 1877, p. 
295. 



and in it act the rollers B^, B^, and the width of face of the 
rollers compels a sliding action, forward on the outer edge and 
backward on the inner. The trough may be stationary and the 
shaft a, carrying the rollers, revolve. Rollers with inclined 
axes are also used for grinding, and a similar device has been 
made for straightening round rods. 




Fig. 564. 

? 19S. 
Roller Bearings. 

Roller bearings, sometimes called anti-friction rollers, maybe 
used in either of two forms : 

((?), in such manner that the rollers are carried in their own 
bearings, the latter receiving the load ; 

(b), or in such a manner that the rollers are placed between 
two moving surfaces and act with a rolling motion upon both of 
, them. 

Roller bearings are used in connection with surfaces which are 
flat, round, or even spiral. Examples of rollers upon cylindri- 
cal surfaces are given in Fig. 565, in which a and b are forms 
used on pillar cranes, and b-^ is the more general form of b. Roll- 
ers are also used in axle bearings, and in heavy pulley blocks, 
where indeed the sheaves themselves are a form of friction roller. 





Fig. 565. 

A form of roller bearing which is subject to very heavy loads 
is that used to carry the ends of bridge beams and trusses, to 
provide for expansion and contraction. These are made either 
with round rollers, as at a, Fig. 566, or with double segments, as 
at 5. 

For round, solid rollers, the load may approximately be in- 
vestigated as follows : — Let / be the length, r the radius of each 
I roller, and P the load. This load will be carried by a surface of 
a width b, included in the angle (measured at the centre of the 
roller) /J = 2?>. We have for the relation of these elements : 



P = Elr 



48 



and S- 



16 



/3^ 



£ being the modulus of elasticity, and 5" the fibre stress upon 
the material. 
Also : 



5=0.83 ^/£: 



^(r 



and 






It will be seen that for any given material the relation - — 
can be so made as to keep the stress within practicable limits. 



THE CONSTRUCTOR. 



127 



These may be chosen as follows, both surfaces being of the 
same material : 

Cast Iron. U^roit^ht Iron. Steei {hardened) 

E^ 14,220,000 28,440,000 42,660,000 

p 

— = 425 to 500 340 to 400 1000 to 1400 

/ r 

5'^ 11,000 to 12,000 11,000 to 13,500 25,000 to 32,000 




- -<-x— X--)(-°-X-°)(-°->---^§ 




Fig. 566. 

Example I. The bridge over the Elbe at Hohnstorf has spans of 330 feet* 
The bearings are made of cast iron of the form shown at /'. The pressure is 
792.000 pounds on six rollers, the dimensions of the latter being, / = 53", 
»-= 4,125". 

We have therefore — ,= — ^ = 603.8, 

rl 53X4.125 

hence p= ^/ i? A /603.8 = 0.126. 

^ 14,220,000 ^ 

This gives for the breadth h of the contact surface under this load. 



S = fi r = 4.125 X 0.126 = 0.522, and 5 = 



14.220,000 



(0.126)2 = 14,280 lbs. 



Example 2. Bridge over the Rhine at Wesel ; span 125.7 feet, rollers and 
bearings of hardened steel. The load is 770,000 pounds ou si.x rollers, as 

P 



shown at a, and /= 27 75" 



: 3.875". These values give 



r / 



and h =0.43", and 5= 32.450 lbs. 

Example s Clifton Bridge at Niagara. The load of 171,600 lbs. is carried 
upon II steel rollers, on bearings of the same material, their dimensions be- 
ing/ =6.3", r=o.6". This gives a high value for -^ — = 4127; )3 = 0.1 7, hence 
d = 0.102, and 5= 74,210 lbs. 

Ball bearings are frequently used instead of cjdindrical bear- 
ings, and for some <brms of journals are most convenient, 
although the bearing surfaces being only points, they are not so 
well adapted for heavy pressures. 




A form of roller bearing used in agricultural machinery is that 
of Cambon, shown in Fig. 567. The steel ring, with semicircu- 
lar groove is secured to the shaft, and in the groove, or globoid 
ring, the steel balls, 09 to 13 in number, are placed. These are 
"held in place by a corresponding external ring, made in halves. 
The outer ring is held in the journal box. Cambon uses balls of 
Jl" to i" in diameter, which are rolled in a mill of similar con- 
struction to the bearing. 

It may be remarked that when roller bearings are used for car 



or wagon wheels we have a combination of friction wheels, since 
the wheels themselves are properly friction rollers and the intro- 
duction of rollers into the axle bearing makes the latter device 
what may be termed friction wheels of the second order. 

Another system of the higher order is the very ingenious ar- 
rangement of planet rollers of Mechwart. In this apparatus, 
which seems to have been so completely conceived by the in- 
ventor as to be incapable of further improvement, friction rollers 
are utilized to the fullest possible extent. The system is indi- 
cated in the diagram Fig. 568. J?, A*,, J?.,, are the rollers of a 
roller-mill. The axis a, of the roller J?, is carried in a station- 
ary bearing ; the axes d and r are carried upon links b, (7, and />., 
a.,, so that they may be moved to or from a, or may be pressed 
against the latter. The length of these links is governed by a 
screw adjustment. The rollers are pressed together by the ring- 
roller 7?,, acting upon the planet roller r, and rollers r, and ^'2, 
the latter bein,g loose upon the axes of the main rolls j?; and /t'j. 
The planet roller r.^ acts a,gainst the roller 7; which is fast on the 
axis of /?. If it is desired to exert greater pressure upon the 
rollers, the roller 7?, is forced towards A'3 by means of the lever 
combination a, 6, f, d^, the lever </, r, being held in position by 
a ratchet section, the position being changed as the rollers wear. 
The ring roller 7?, reduces very greatly the wear upon the jour- 
nals of the grinding rollers, as it converts the greater part of 
their journal friction into rolling friction. In order to equalize 
the effect of the weight of the upper roller, the lower roller 7?, 
is counterbalanced by a weight, which, acting through the sys- 
tem of levers c., d., a^ exerts an upward pressure equal to the 
combined weight of 7?, and /t'2. 




Fig. 568. 

The above described apparatus is fitted to both ends of the 
rollers. In order to provide for an}' slight inequality in diame- 
ter between the opposite ends of the rollers, another adjustment 
is provided. This consists of the lever a d f,to which the planet 
roller is suspended by the link b a^. This permits the planet 
roller to be forced into the narrower space between r and 7?,, by 
means of the worm and worm sector shown at c. The ring 7?3 
is a continuous steel forging, and the rollers are chilled castings. 
The rollers 7?, 7?,, 7?.^ are geared together, the gears having 
double spiral teeth, as shown in J 222. 

Friction rollers are sometimes used in connection with toothed 
gearing, an example of which will be seen later in Fig. 5S9. 
Whitworth used rollers in the place of a nut in his screw planing 
machine, and they are used in worm gearing by Bourdon, and 
the higher form of worm, the globoid (see i 224) by Jensen, and 
b}' Hawkins. Many applications are also found among instru- 
ments of precision, notably Atwood's Machine, and Amsler's 
Planimeter. 



CHAPTER XVII. 
TOOTHED GEARING. 

I 199- 

Classification of Ge-ar Wheew. 

The relative position of the axes of gear wheels governs their 
general form, although not to so great an extent as in the case 
of friction wheels. This is due to the fact that the geometric 
shapes, which in the case of friction wheels form the actual 
surfaces, are only theoretically used in the case of gear wheels 
as forms upon which to design the teeth. 

Gear wheels for parallel axes are called spur gears ; their form 
is based on the cylinder. Wheels for inclined intersecting axes 
are based upon cones, and are termed conical gears, or more 
commonly bevel gears. For inclined, non-intersecting axes, the 



128 



THE CONSTRUCTOR. 



base form is the hyperboloid, which name is also given to the 
gears. For many applications of inclined axes the teeth are 
made spiral, giving the variotis forms of spiral gears and worm 
gears. 

If the motion is to be transmitted at a uniform rate, the base 
figures are solids of revolution (cylinders, cones, hyperboloids) ; 
the wheels themselves being round, while if the motion is not 
to be transmitted uniformly the outlines will be irregular. In 
the following discussion ouly round gear wheels will be con- 
sidered. 

A. THE CONSTRUCTION OF SPUR TEETH. 



Accuracy in spacing is of especial importance in the change 
gears of a lathe, as any error in a gear produces a corresponding 
defect in the screw which is being cut. Such defects are still 
more apparent if the lathe is used for cutting spiral gears (see 
^221, below). The smooth motion which such spiral gears are 
intended to produce may thus be p^'evented by irregular cutting. 
The choice of tooth outline to be adopted, either for the entire 
product of an establishment, or for any class of work, can only 
be made after a careful consideration of all the conditions upon 
which so much depends. These considerations will be taken 
up and discussed in the following sections : 



GENER.ii, Considerations. 

The form of gear teeth may be so chosen that all gears of the 
same pitch will work together. Wheels of this sort are called 
intex-changeable, while wheels which are not so made will run 
only in pairs. 

In each pair of round wheels there are two circles, struck 
from the centres of the wheels, which have at each moment the 
same linear velocity, and are called in general the ratio circles. 
The particular ratio circles for a pair of spur gear wheels are 
called their pitch circles. Upon these circumferences are laid 
out the pitch divisions, i. e., the spacings from centre to centre 
of the teeth. 

The teeth themselves are prismatic in shape, the base of the 
prism being the outline of the tooth. The portion of the tooth 
which projects beyond the pitch cylinder is called the point of 
the tooth, and that portion within the pitch cylinder is the base. 
The surfaces of the point are called the faces of the tooth, and 
the surfaces of the base, the flanks. 




Fig. 569. 

In spur gear teeth we also have, Fig. 569, the length /, the 
breadth on face b, the tooth thickness rf, the pitch being indi- 
cated by /, the two latter being measured on the curve of the 
pitch circle. 

All the teeth in one and the same wheel are made of the same 
thickness and same spacing, so that any tooth will fit into any 
space. It follows from this that, the spaces being made of 
suitable size to receive the teeth, that the inverse ratio of the 
number of revolutions « and n-^ of a pair of wheels is equal to the 
direct ratio of the respective numbers of teeth Z, and Z i, or: 

This statement is equally true for circular and non-circular 
wheels. It also holds good if the thickness of the teeth is dif- 
ferent at different portions of the circumference, providing only 
that care is taken that the spaces in the smaller gear come 
around to meet their proper teeth each revolution. If, there- 
fore, under these conditions we have the number of teeth given 
for any case, we may consider the above relation as l/ie funda- 
mental forinula of tlie tnuismission of motion by toothed gear- 
ing. This is rather to be considered as an inevitable principle 
of construction rather than a fine geometrical distinction. It 
depends upon the primitive form of gear construction which has 
been in use for centuries in the Orient, where no other care is 
taken in the proportioning of gears except that they are large 
enough and that the pin teeth are sufficiently strong. 

No general principle can be laid down for the form of the 
flanks of teeth. For roinid wheels, the ratio of the angular 
velocities, /'. e., that of the differeutials of the simultaneous 

2, 
angles of rotation u-^ and u must equal the ratio — . This affects 

,, ,. d u, 

the ratio ' 

aw 
depends. 

The form of the teeth is of great importance. Especially ne- 
cessary is it that the division of the pitch shall be accurately 
made ; errors in the shape of the flanks are even less injurious 
than errors in the dividing. Accurate spacing can only be ac- 
complished by the use of suitable gear-cutting machinery, and 
such machines are now in general use.* 

* The machines most extensively in use in Germany are those of the Ber- 
lin .^iihalt Maschinenfabrik, in Berlin, and the Maschinenbauanstalt of 
Briegleb, Hansen & Ca, in Gotha. 



the flanks as being those surfaces upon which 



« 201. 

Pitch Radius. Circumferentiai, Division. 

For any pitch t, and number of teeth Z for a round wheel we 
have for the radius R of the pitch circle : 



R_ 

t 



Z 



= 0.15916 z . 



which : 



;ives, according to formula (186) 
R «, 
R 



_ "I 



(187) 



(188) 



The radius obtained from formula (1S7) is never a whole num- 
ber, because tt is an irrational number, so that R will always 
contain a fraction if the pitch is a whole number. The follow- 
ing table will facilitate the computation in such cases. If the 
irrational feature is to be kept out of the value of R, the length 
of the pitch divisions must not be made whole numbers, but 
fractions or multiples of tt, and this method is used in many es- 
tablishments. If we call the pitch = t, we have under this plan : 



R = 



Z r t 






(189) 



This corresponds to the so-called " diametrical pitch " system 
of England and America. 

Example. Suppose a wheel of 24 teeth and a pitch of 6 X 3.14 millimetres, 

24 
we have according to (iSy), for the radius i?, of its pitch circle, R = — X 6= 

2 
72 mm, and if we have in English units a pitch of ^q X 3.14 for a wheel of 30 

teeth, we have according to (18 



,^=-T-X3 = 



45 
16 



13 
'16 



A convenient instrument in this connection is a circumfer- 
ence scale. This consists of a prismatic rule of wood or metal 
upon which, for the metric system, a length of 314 millimetres 
is laid off, and on a parallel line the same distance is divided 
into 100 equal parts. Corresponding points on the two scales 
will then have to each other the ratio -i : ^r. This scale is also 
useful for the rectification of circles and circular arcs. Similar 
scales may be prepared upon sixteenths, tenths or any subdivi- 
sion of the inch. 

In the following discussion both methods will be used, namely : 
that in which the pitch is taken in rational numbers, thus mak- 
ing the radius irrational ; and that in which the pitch is made 
rational in units of the circumference scale, and hence the radius 
becomes rational. The following table is not to be confounded 
with that of Doukin, * made according to the expression, 



F) 



which gives the radius of the circumscribing circle of a regular 
polygon of Z sides, each having a length equal to t. This latter 
radius differs from the radius R above referred to for small 
values of Z, and confusion in this respect has given rise to 
numerous errors. 



Tabi,e of Radii op Pitch Circles. 

Examples in the use of the following table. (Note. This 
table was calculated for use with the metric system, in which 
the pitch is generally taken in millimetres. It may, however, 
be used equally well in English units, by taking the pitch in 
sixteenths, in order to make the divisions sufficiently small.) 

Example I. A wheel of 63 teeth, and i^" pitch is to be made; required, the 
radius of the pitch circle. The pitch is here 30 sixteenths, and w^a have at 
the intersection of the columns for 60 and 3, the number 10.03, hence R = 
10.03 X ^ = 10.03 X 30 = 300.9 sixteenths or 18.8" giving a diameter of 37.6''. 

The table may ako be used to determine the number of teeth 
when the pitch is chosen and the radius given. 



* See Salzenburg's Vortiige, p 93, and others. 



THE CONSTRUCTOR. 



129 



Exavipie 2. Given a wheel of 40 inches radius and 1.6" pitch. This gives 
— = - -- = 25. The nearest value to this in the table is 24.99^; at the inter- 

/ ID 

section of 150, and 7, and hence 157 is the number of teeth. 

When the radius aud number of teeth are given the table 
may be used to find the pitch. 

Example 3. Given R = 15:^" Z = 54. "We find in the table at the intersec- 
tion of 50 and 4, the value of = 8.59. We then have i = = 1.S3". 

t S.59 



z 





I 


2 


3 


4 


5 


6 


7 


S 


9 





0.00 


0-159 


0.318 


0.477 


0.637 


0.796 


0-955 


1. 114 


.1.273 


1-432 


10 


J-59 


'-75 


1.91 


2.07 


2.23 


2-39 


2-55 


2.71 


2.86 


3.02 


20 


3-18 


3-34 


3-50 


3.66 


3-S2 


3-98 


4.14 


4-30 


4.46 


4.62 


30 


4-77 


4-93 


.5.09 


5-25 


5-41 


5-57 


5-73 


5 -89 


6.05 


6.21 


40 


6.37 


6.53 


6.68 


6.84 


7.00 


7.16 


7-32 


7.4S 


7-64 


7.80 


so 


7.96 


8.12 


8.28 


8.44 


8-59 


8.75 


8.91 


9.07 


9-23 


9-39 


60 


9-55 


9.71 


9.87 


10.03 


10.19 


10.35 


10.50 


10.66 


10.82 


10.98 


70 


11.14 


11.30 


11.46 


11.62 


11.78 


11.94 


12.10 


12.25 


12.41 


12.57 


80 


12.73 


12.S9 


13-05 


13.21 


'3-37 


13.53 


13.69 


13.85 


14.01 


14.16 


90 


14-32 


14.48 


14.64 


1 4. So 


14.96 


15.12 


15.28 


15.44, 15.60 


15-76 


100 


15-92 


16.07 


16.23 


16.39 


16.55 


16.71 


16.87 


17-03 


17.19 


17-35 


no 


17.51 


17.67 


17.83 


17.9S 


1S.14 


18.30 


18.46 


18.62 


18.78 


18.94 


120 


19.10 


19.26 


19.42 


19.58 


19.73 


19.89 


20.05 


20.21 


20.37 


20.53 


130 


20.69 


20.85 


21.01 


21.17 


21-33 


21.49 


21.65 


2 [.80 


21.96 


22.12 


140 


22.28 


22.44 


22.60 


22.76 


22.92 


23.08 


23.24 


23.40 


23-55 


23.71 


150 


23-87 


24.03 


24.19 


24.35 


24-51 


24.67 


24-S3 


24-99 


25-15 


25-31 


160 


25-46 


25.62 


25.78 


25.94 


26.10 


26.26 


26.42 


26.58 


26.74 


26.90 


170 


27.06 


27.21 


27.37 


27.53 


27.69 


27-85 


28.01 


2S.17j2S.33 


28.49 


I So 


28.65 


28.81 


28.97 


29.13 


29.28 


29.44 


29.60 


29.761 29.92 


30.08 


190 


30.24 


30.40 


30.56 


30.72 


30.88 


31-04 


31-19 


31-35 


3i-5> 


31.67 


200 


3 1 -S3 


31-99 


32.15 


32-31 


32-47 


32.63 


32-79 


32-95 


33-10 


33.26 


210 


33-42 


33-58 


33.74 


33.90 


34.06 


34.22 


34-38 


34.54 


34.70 


34.85 


220 


35-01 


35-17 


35.33 


35-49 


35.65 


35-81 


35-97 


36.13 


36.29 


36.45 


230 


36.61 


36.76 


36.92 


37.08 


37.24 


37-40 


37-56 


37-72 


37-88 


38.04 


240 


38.20 


38.36 


38.51 


38.67 


38-S3 


3S.99 


39-15 


39-31 


39.47 


39.63 


250 


39-79 


39,95 


40.11 


40.27 


40.42 


40.58 


40.74 


40.90 


41.06 


41.22 


260 


41.38 


41.54 


41.70 


41.86 


42.02 


42. iS 


42.34 


42.49 


42.65 


42.81 


270 


42.97 


43-13 


43.29 


43.45 


43.61 


43-77 


43-93 


44.09 


44.25 


44.40 


280 


44.56 


44-72 


44-88 


45-04 


45.20 


45-36 


45-52 


45-68 


45.84 


46.00 


290 


46.1s 


46.31 


46.47 


46.63 


46.79 


46.95 


47-11 


47.27 


47-43 


47-59 



? 203. 
Generai, Solution of Tooth Outunes. 

In a pair of gear wheels, the two tooth outlines which work 
together lie in a section at right angles to the axes of the wheels 
and in the plane of this section the construction and action of 
the teeth is to be considered. The so-called general solution of 
tooth outlines is that by which, if a form of tooth be given for 
one wheel, the proper form of tooth for the other wheel may be 
drawn so that the motion will be transmitted with a uniform 
velocity ratio. Several such solutions will be given. 

I. The Author's First Solution. Fig. 570. Given the tooth 





Fig. 570. 



Fig. 571. 



profile a S b c, also the pitch circle T, of the wheel O, and the 
pitch circle 7", of the wheel Oj ; required the tooth curve a^ S 
for the wheel Oj. 

Place the given curve so that the point S, where it crosses the 
pitch circle, lies on the line joining the centres O O,, thus mak- 
ing 6", a point common to both profiles. In order to find a 
second point a^, which shall work in contact with a point a, 



draw a 1 normal to the given curve at a, make the arc S i' = 
arc 5' I, and the distance i s^ ^ S l', and S s-^ = 1' i. Then 
with ^as a centre strike an arc with a radius S, (/, and from i', 
an arc with a radius i a, and the intersection of these arcs will 
be the desired point a^ of the required curve. For such points 
as <-, where the normal to the curve does not intersect the 
pitch circle the given pitch circles cannot be used, therefore if 
these points are required the pitch circles must be transposed 
(exaggerated in the figure). The curve thus found sometimes 
assumes an impracticable form without being geometrically 
incorrect. 

II. Abridged Solution. [Poncelet.) Fig. 571. Mark off ou 

the pitch circle T^, the points s^, t^, n^t'i , which 

roll into contact with points j', t, u, v, . ... of the given 

circle T, draw from s^, t-^, «i, &c., arcs with radii respectively 
equal in length to the normals to the given tooth outline va, u 
c, etc., then will a curve drawn tangent to these arcs be the re- 
quired outline. The points s, t, u, v, should be taken close to- 
gether. If the lengths of the normals va, u c, etc., are taken 
backward from the points 5j, /j, //,, &c., instead of forward, the 
outline for an internal gear tooth will be obtained for the wheel 




Fig. 573- 



III. The Author's Second Solution. Fig. 572, The tooth out- 
line a 5 c Srf <? is given, and its pitch circle T, also the pitch 
circle T^. Draw the normals a i, b 2, c 3, &c., also draw from O, 
as a centre, arcs through a, b, c, &c., and make S 1 = ai, .SII = 
b2, Slll = cT„ &.C., and draw the curve I, II, 111,5, IV, V, &c. ; 
this curve will be the path of the point of contact of the teeth, 
and may be called the Line of Action. 

IV. Theoretical Profile of the Flank. Fig. 573. lu order to 
obtain the necessary strength it is frequently desirable to make 
the root of the tooth as thick as can be done without interfering 
with the path of the face of the corresponding tooth of the other 
gear. This path ma5' be determined in the following manner. 
Let a S b\>^ the profile of the tooth for the wheel T, (7, 5, b^ that 
for the wheel 7",, a^ a^ the prolongation of the flank outline for 
the latter tooth, and I S II the line of action between the limits 
of the outside diameter circles K aud Ky Lay off from S, on 
both pitch circles the corresponding spaces 5 i, i 2, 2 3, &c., 5" 
i', i' 2', 2' 3', &c. Take in the dividers successively S a, 1 a, 2 
a, 3 a, &c., and describe arcs from i', 2', 3', &c., and the envel- 
ope of these arcs will give the path (7 n-j g, or so-called theoreti- 
cal profile of the flank. The actual profile of the flank a-^f is 
drawn tangeut to the theoretical curve, to the point where it 
crosses the clearance circle F^. The theoretical curve is a pro- 
longed or abridged C3'cloidal curve (see ? 205). In the figure, in 
which 7" is a straight line, or rack, the curve is an abridged 
evolute. 

^204. 

The Action of Ge-^r Teeth. 

In solution III, of the preceding section, reference was made 
to the li?ie of action * of a pair of gear wheels, and this line 
bears an important relation to the theory of the action of gear 
wheels. 

The line of action intersects the pitch circle at the same point 
as the tooth profile and cuts the latter at right angles, so that 
the tangent TV iV of the line of action (Fig. 572) is normal to the 
tooth profile. Each point of action corresponds to a point of 
contact of the teeth and also to a point of contact of each of the 
pitch circles ; so that, for example, the point II of the line of ac- 
tion corresponds to the point 2 on T, and 2' on T'. That por- 
tion of the pitch circle between the pitch point of the line of 



* First discussed in Moll & Reuleaux's 
schineubau." 



Konstruktionslehre fiir den Ma- 



130 



THE CONSTRUCTOR. 



actiou aud the initial poiut of contact is called the rolling arc 
for the given point. For example 5' 2 is the rolling arc on T', 
for the point II, and S ^' on 7'j, for the same point. 

The sum of the rolling arcs between the two extreme points 
(arc I 5-1- 55, or arc i' 5 -|- .S5') is called the arc of actiou, 
and its length indicates the duration of the action of the given 
pair of teeth, which is easily determined graphically. It depends 
upon the length of that portion of the line of action which it is 
desired to use. This is usually taken between the limits of the 
circles of the outside and the base of the teeth, which gives in 
Fig. 572 the line of action VI. 

For any wheel of given tooth outline and pitch diameter there 
is but one line of action, and lor a given line of action but one 
tooth profile. This latter can only be determined from the line 
of action when the rolling arcs for the pitch poiut of the Hue of 
action are also given. 

For cycloidal teeth the rolling arc is also the line of actiou 
aud for this reason the geometrical discussion is much simpli- 
fied. In order that a pair of gear wheels should work properly 
together, their lines of action should correspond and their roll- 
ing arcs be of equal length for homologous points of action. 
By conforming to these conditions anj' number of gear wheels 
miy be made to operate with a given wheel. Such wheels are 
said to be interchangeable or series wheels, since the common 
line of actiou is symmetrically disposed on each side of the pitch 
circle, as well as on each side of a radial line passing through 
its pitch point. 

The ray drawn from the pitch point through any point on the 
line of action (as SI, in i'ig. 572) gives the direction of the 
pressure between the teeth for that point. 




\ 205- 
The Cyci<oidai, Curves. 

For the generation of tooth outlines for gears to be used in- 
terchangeably in series, the cycloidal curves or those produced 
by rolling circles are the best. When one circle rolls upou 
another in the same plane without sliding, each point in any 
radius describes a curve which is called either a common, ex- 
tended or abridged cycloid, according as the point is situated 
on the circumference of the circle, or on a radial line without or 
within the circumference. 

The stationary circle is the base circle of the curve, and its 
radius will be here indicated by 7?; and the radius of the rolling 
circle by r. If we consider either radius negative when it lies 
"within the other circle, and negative when it lies without, we 
may distinguish the five kinds of cycloidal curves whose radii 
have the relation R, and r, in the following manner : 



Base Circle. 


RoIHug- Circle. 


Corresponding Curve Name. 


-f-co 

— R 
■\- R 
+ R 


■\-r 
4- r 

± CO 


Epicycloid. 
Orthocycloid.* 
Hypocycloid. 
Evolute of circle. 
Pericycloid. 



The following properties are common to all five curves : 

1. The normal to any element of the curve passes through the 
corresponding point of contact of the generating and base circles. 

2. The centre of curvature of any element of the curve is at 
the intersection of the normal with a right line whicli joins the 
starting point of the curve ivith the centre of the base circle. 
For the extended, or abridged cycloidal curves the starting point 
is taken on the radius prolonged of the curve element, at right 
angles to a normal to the curve at the poiut of contact of the 
rolling circles. 



Upon the first property depends the suitability of the cycloidal 
curves for use as tooth outlines, and in the second lies the prac- 
ticability of approximating them by circular arcs. 

I 206. 
The Generation of Cyci,oidai, Curves. 

I. Exact Solution. Fig. 574. C is the base circle, ?r the roll- 
ing circle, A the starting point of the curve. Lay off from A, 
on G and W, small arcs of uniform spacing, and let a, and (Tj 
be two of the corresponding points of division. From A, with 
radius a, «,, strike an arc, aud from a, with the chord A flj, 
another arc, and the intersection of the two arcs at P, will be a 
point in the curve. 

This solution, which is shown in Fig. 574, both for external 
and internal rolling, holds good for all five curves. 




IT. Abridged Solution. From the points i, 2, 3, . . . a, with 
radii equal to the corresponding chords of the rolling circle, 
strike arcs, which arcs will include the entire curve with suffi- 
cient accuracy if the points of division be taken sufficiently 
close together. 

In order to draw the extended or abridged curve, starting say 
at B, determine first a point/', on the ordinarj' curve, then draw 
from a, with a radius a.^ B, an arc, and from P, another with 
radius A B , apd the two arcs will intersect in a poiut O, of the 
curve. 

Or, draw through a-^ a radius (7., b in the rolling circle and 
through b, an arc b C, concentric with the base circle, and make 
a., Oi = A b, then will (9, be the point in the curve for the roll- 
iiig'of the arc A a^ upon A a.,. 




Fig. 576. 



207. 



*The author gives this name to the comnion cycloid because the latter 
term properly includes the whole class. 



The Generation of Interchangeabi,e Teeth. 

The tooth profile for interchangeable gears is generated in a 
similar manner, both for external and internal gears, by using a 
rolling circle of constant diameter for each pitch. 

I. External Teeth. Fig. S75. Given the number of teeth .^, 

i ' Z t 

and pitch /, or ratio — of the wheel. Make O S = R=^ 

■K 2 -n- 

= — Z( — 1, and the radius r^ of the rolling circle lF^o.Sy5i 
2 \ ^ I 



THE CONSTRUCTOR. 



131 



tr =: 2.75 — ; draw the outside circle of the teeth A', with a ra- 
dius ^ J? + °-3 A ^"id the inside circle F, with radius =^ R — 

0.4 t, and make the thickness of tooth ^ — A Arc Sd = arc a b ; 

40 
arc S c = arc i c. S a, the face curve, is generated by the roll- 
ing of jy upon 7 ; S i, the flank curve, by the rolling of IV in 
T. For pinions of eleven teeth, S i becomes a straight line and 
radial. Pinions with as few as seven teeth can be made to work 
on this system, for although the flanks are undercut, they are 
still within the limits of the theoretical flank profile (see | 203, 
and Fig. 573, where a seven tooth pinion is shown with a rack 
tooth). The backlash is yV i. 

II. Internal Teeth. Fig. 576. The generation of internal 
teeth is similar to the preceding. The radius of base circle is 
— R, and the length of tooth above and below the pitch circle 

is 0.3 /, and 0.4 /, as before ; 7\ = 0.S75 /* ^ 2.75 — , and the thick- 

TT 

ness of tooth ^~t. The flank S ais generated by rolling IF 

40 . 

upon T, and the face S J, by rolling ?f^ inside of 7. 

In the case of a rack R = x , S a and 6' / then become similar 
portions of the common orthocycloid (see Fig. 573). 

In teeth of this form the line of action coincides with the 
rolling circles, the portion included being = arc 6 a -{- the cor- 
responding arc ^1 «! of the opposing wheel, when both are ex- 
ternal gears, and-}- the arc c i for an internal gear working with 
a spur gear. The duration of action e, varies between 1.22 and 
1.60. 

?20S. 

Tooth Outi,ines of CiRcur,.\R Arcs. 

Instead of using the exact tooth outlines as generated by the 
rolling circles, two circular arcs may be used as a close approx- 
imation (see \ 205). 



£.raOT//c I.— Given ^• — es, ^= 1.3125", we have forthe radius fortheface 
of the teeth : 




Fig. 577. 

Fig. 577. Draw the pitch circle T, and outer and inner circles 
/if and F, as before, also the centres BI, and iJ/j of the rolling 
circles fKand /f,, which latter are in contact with each other 
and with the pitch circle at 6". Draw the diameters B 1\[ D and 
By Ml D-i in such manner that the angle B RI S =^ angle B^ yl/j 
Si = 30°, join B and Bi with the prolonged line C^ B S By, and 
draw through D and Di the lines O D and O D^C^\ then will 
the intersections at C and C, with Bi C S C^ be the required 
centres of curvature for the arcs a B b, and c Bi 2. 

Through Cand C^ draw circles with O as a centre, and on 
these circles the centres for all the teeth will be found, the arc 
a B b being struck from C, and c B^ i from C^. 

The radii of curvature p may be calculated from the following 
ormula : 



p 1 Z 

-f = 0.45 — 7~- — 
t Z ± 11 



— and 



(i) 



= 1.42- 



2Z: 



. . . (190) 



The plus sign gives the radius C B, for the face {pa) and the 
minus sign gives the flank radius C^B^, {pi). The flank should 
be joined to the bottom of the space by a small circular fillet. 



p^=i'3i25X 0.4s 
For the flank radius we have; 



126 -f II 
63 + II 



P:' 



1.3125 X 0.45 



EI26- 



Exaiiiple 2. — Giveu Z= 11, — = 
p^j = °-4X 1.42 



«3- 
0.4. We have: 

=- + " .<,.// 



= 1.306". 



Also, 



= 0.4 X 1.42 



hence the flanks are straight radial lines. 
Example 3.— Given Z—-],t = i". We have : 

14 -I- It 



Also, 



^ 3 X 0.4 
2X0.4 



7 + II 
14 — II 



The negative sign indicates the undercut flank. This is shown in Fig. 573 

It is better to use the exact method given in \ 207, for wheels 
with fewer than fifteen teeth, as the approximation becomes less 
accurate for the lower numbers. 

I 209. 

Evoi,uTE Teeth for Interchangeabi^e Gears. 

Gear teeth may be given the evolute form, which cur^'e is de- 
veloped by unwrapping a line from a base circle, which is con- 
centric with, and bears a definite relation to the pitch circle. 





Fig. 57S. 






Fig. 579- 



External and Internal Teeth. Fig. 57S and Fig. 579. Given 
the number of teeth Z, and pitch /, or ratio — for the required 

wheel. Make O S = R ^ -^ — = — Z(^) and draw the 

27r 2 \^ y 

outer and inner circles, giving the distances_/^ 0.4 /, /t ^ 0.3 i 
above and below the pitch circle, also make the thickness of the 

tooth = i5 /. 
40 
Draw the line iV S A\ at an angle of 75° with O S, and it will 




be tangent to the base circle G, the radius of which = r = 0.966 
R = 0.154 Z i, = 0.483 Zl — J. If now we unwrap the line iV 

.Supon the circle G, from 5' outward to a, and inward to^, the 
path a Sg' of the point 5' will be the required tooth outline, 
which for wheels of fewer than 55 teeth may be prolonged by a 
radial line to reach the bottom circle. 

The line of action is the straight line N Ni ; and extends from 
S b to S bi on the other gear, or in the internal gear to 5 c. To 
determine the duration of contact e the pitch / can be carried to 



132 



THE CONSTRUCTOR. 



. the base circle by drawing radii, and the length measured. 
For two equal wheels of 14 teeth, e is only a little greater than 
unity ; it varies between i and 2.5. 

Rack Teeth. Fig. 580. The profile a S i\z straight and makes 
an angle of 75° with the pitch line T. The angle 75° can readily 
be laid off by using the drawing triangles of 45° and 30° to- 
gether. 

For low numbered pinions the base circle closel}' approaches 
the pitch circle. This sometimes introduces an error into the 




action. If the portion S B, of the line N N^, which lies be- 
tween the pitch and base circles. Fig. 5S1, is shorter than the 
length of face of the opposing tooth, the point a will interfere 
■with the flank of the pinion tooth, as shown in the path afg. 
(See also Fig. 573.) In order to avoid this, the tooth to which 
the point a belongs must not extend above the line K' K' . 

This exists for teeth made in the manner given, when Z ^28. 
Another method of avoiding this difficulty is to round off the 
tooth at (Z, and this is more frequently adopted in practice. An 
important application of evolute teeth is shown in \ 222. 

§210. 

Pin Teeth. 

Teeth with radial flanks can always be generated by making 
the inner rolling circle for each wheel equal in diameter to one- 
half the pitch circle. This will give radial flanks, and curved 




Fig. 582. 



faces to both gears, but wheels made on this system are not in- 
terchangeable, and are therefore not practical for general ma- 
chine construction. Such teeth are still much used by watch- 
makers on account of the ease with which they may be fitted 
by filing. 

If the diameter of the rolling circle is made greater than the 
radius of the pitch circle a form of tooth is obtained which is 
practicable, but which is comparatively little used. 

If, in a single pair of wheels, the rolling circle be taken for 
one wheel equal to the pitch circle, of the other wheel, we obtain 
for the teeth of the wheel upon which the rolling is done, an 
outline of cycloidal form, while the teeth of the other wheel be- 
come mere points. In practice these points are the centres 
about which pins are described and such gears are called pin- 
tooth gears. 

External Pin-tooth Gearing. Fig. 5S2. The pins are circular 

in section and in diameter equal to — ;",• the tooth profile for 

40 
the wheel R^ is then a curve parallel to the path 6" a, described 
by rolling the circle T'ou T",. The arc S b ^ab, and circles of 
the diameter of the pin, struck from successive points of the 
path 5 a, will outline the tooth profile c d, the flank d i being a 



circular quadrant. The curve of action S I \& limited by the 
outer circle K^ at /, and is in all cases greater than t, generally 
not less than i.i A This gives the limit of tooth length k^f and 
also determines ky If it is desired to construct the actual line 
of action, the method of case III, ? 203, may be employed. 
Fig. 583 shows a pinion of six pins gearing into a wheel of 

24 teeth. The diameter of the pins is here made = — The 

. . 3 

flanks of the 24 tooth wheel are made radial with square cor- 
ners in order to permit ready filing and finishing. 




Fig. 5S4. 



Fig. 5S5. 



Internal Pin-tooth Gearing. Fig. 5S4. This is similar to the 
preceding. The tooth profile c rf is a parallel to the curve 5' z, 
generated by rolling /'in T^ the arc S b = i b. S I is the line 
of action and is made equal to, or greater than i.i /. The flank 
d a\s made radial. 

In Fig. 585 the pinion is made with the pin teeth and the spur 
teeth are on the internal gear. The profile c d \s, parallel to the 
curve S a, generated by rolling 7"upon T^\ the arc S b = a b, 
S / is the line of action, as above, and is made equal to, or 
greater than i.i t ; the flank d i is made radial. 

If in Fig. 5S4 we make the radius 7?! infinitely great, we ob- 
tain a rack, and the tooth profile is a curve parallel to the com- 
mon cycloid. If we make R, in Fig. 585, infinitely great, we 
obtain a common form of rack, with pin teeth. 

Pin teeth have the practical advantage that they may readily 
be turned in the lathe. They are especially adapted for situa- 
tions where they are exposed to the weather, as in sluices, swing 
bridges, wind-tnills, etc. In such cases the pins are often made 
of round bar iron, without being turned. 




Fig. 5S6. 

Double Pin Gearing. Fig. 5S6. If two gears on this system 
are run together, one gear may be made with very few teeth, 
and hence a great difference in velocity ratio obtained, with a 
minimum distance between centres. In thi? case both pitch 
circles become rolling circles. S a, the pinion face, is generated 
by rolling 7; on T, the action extending on S I for the point S 
on the wheel T. S «i, the gear tooth face, is generated by roll- 
ing 7" on 7",, the action extending on the line S If, for the point 
S, on the wheel T^ S i, the flank profile, is made to conform 
to the theoretical profile .S'«, o', (see case IV, ? 203), and the 
other flank is made in a similar manner from the theoretical 
profile Sag. Such gears are sometimes used in hoisting ma- 
chinery. 



THE CONSTRUCTOR. 



133 



Disc Wheei^s with Pin Teeth._ 

It is not an essential requirement that the tooth profile shall 
be in the immediate line of the pitch circles, as it can be placed 
within or without to a greater or less extent. In such cases a 
tooth system is obtained in which the teeth of one wheel pass 
almost or entirely around those of the other wheel, and hence 
there can be no so-called bottom circle to the latter teeth. Such 
wheels are so constructed that the teeth are placed upon the 
side or face of a disc, or shield, and are called disc wheels, or 
"shield gearing." * 




Fig. 5S7. 



Fig. 58S. 



For such wheels pin teeth are well adapted. Fig. 5S7 shows 
a pair of such wheels arranged for external action, and Fig. 5S8 
for internal action. One wheel of each pair is fitted with round 
pin teeth, and the other has, in the first case, a tooth profile 
parallel to an extended epicycloid, and in the second case par- 
allel to an extended hypocycloid. 

A peculiar form of disc gearing is shown in Fig. 5S9. In this 
case R = }i j?,, Z ^2, Z^=--/^. the round pins being on R. The 
flanks of R^^ are entirely within the pitch circle, and become 
straight lines parallel to the straight line hj-pocycloid S i. The 
arc of action is about 2 t, and the backlash can be reduced al- 
most to zero, the teeth on R being made as rollers. 




Fig. 5S9. 

If the distance between centres C C?i of a pair of wheels for 
internal action remains constant, and the radius is increased, 
they will overlap entirely, and the pitch circles will cease to ap- 
pear as an element in the construction. The wheels will have 
equal angular velocity and revolve in the same direction. 

Such a pair of disc wheels is shown in Fig. 590. Both wheels 
are made with pin roller teeth, the sum of the pin radii being 
equal to the distance O Oy The pins are shown of equal diam- 
eters, although they may be unequal, as shown in the dotted 
lines. Such wheels may be called Parallel Gears, as two radii 
which are parallel in one position remain parallel at all times.t 

A second form of parallel gears is shown in Fig. 591. The 
curve a b c\% a. circular arc, of radius d a, which includes four 
segments of the lenticular shaped pins for the wheel O^. 

If the pair of parallel gears of Fig. 590 are placed on opposite 
sides of an axis A A^ normal to two adjoining pins and parallel 
to O (9,, the action of the wheels will be correct. In Fig. 592 is 
shown such a pair of right angle wheels. 



Such gear wheels have been described more than once,* but 
are rarely used ; they are well adapted to transmit motion to the 
hands of large tower clocks. 




Fig. 590. 



Fig. 591. 



Mixed Tooth Outunes. Thumb Teeth. 

By combining the preceding forms of teeth, practical shapes 
may often be made for special service. The two following ex- 
amples will illustrate : 

Mixed Outline. Fig. 593. For the low numbered pinions 
sometimes used in hoisting machinery, it is important that the 




Fig. 592. 

pinion teeth shall not be too much undercut, so as to avoid dif- 
ficulty in making the gears. It is desirable that the flanks on 
the pinion should be radial. In order to obtain sufficient dura- 
tion of action, which for a three tooth pinion should not be less 
than 1. 15 t, the face curves of the teeth should be prolonged 




Fig. 593- 

until they intersect. The curve 6' « is an arc of an evolute 
formed by unwrapping the pitch line TJ from the circle T ; S i 
is the radial flank, obtained by rolling the circle /r of radius = 
Yz R in T; S (7, g-^ is the theoretical profile for the tooth space 
for the wheel T. 

S a acts with the point 5' of the rack tooth over the path S II. 
.S (?i is a cycloidal curve generated by rolling W on 7"i, and acts 
over the path 6' / with the flank 5 i of the wheel T. 



* CM^d Scui£i Deniaii in Zonca's Teatro di Machine, Padua, 1621. 
t This form of gearing was described and named by the author in Berlin 
Verhandlung, 1875, p. 294. 



* See Tom Richards' Aide-memoire. 184S, I, p. 656. Willis' Principles of 
Mechanism, "^851, p. 145, Laboulaye, Cinematique, 1854, p. 275. 



134 



THE CONSTRUCTOR. 




Fig. 594- 

Thumb-shaped Teeth. By combining the evolute and epicy- 
cloid, using the two curves for opposite sides of the same tooth 
a profile of great strength is obtained. This form is of especial 
service for heavy driving when the motion is constantly in the 
same direction.* From the peculiar form these have been called 
thumb-shaped teeth. The following proportions will be found 
suitable for cases in ordinary practice. 




Fig. 595- 

Fig. 594. Spur Gearing with Tliuiiib-shapcd Teeth, a S i a.nA 
«! .S /'i are profiles formed of epicycloidal curves, according to 

the description in ^ 207, in which r^ = 0.S75 t or 2.75 — . 

a' S' i' and a-/ 5/ /j are evolute curves developed from base 
circles with radii r' = o.S R, and i\' = 0.8 7?,, giving an angle 
of 53° (more accurately 53° 8'). For wheels of less than fifteen 
teeth, as in the seven toothed pinion shown in Fig. 594, the 
flanks must be modified as shown in ? 203, to avoid interference. 

In Fig. 595 is shown a four-toothed pinion on this system, 
working with a rack. S a and S i-^ are made as before with r^ 
= 0.S75 t and 6' i and 5' «i with r ^= j{ R \ the evolute curves 
being generated as before with an angle of 53°. 



* This form of mixed outline has been described by WiUis in 1851 ; it was 
revived by Gee in 1S76 and used in practice ; he made the angle a greater 
than here given, viz. 68°. 



The rack teeth are made straight on the one side, as already 
shown for rack teeth on the evolute system. Applications for 
teeth of this form are given in § 226. 

I 213- 

Tooth Friction in Spur Gearing. 

The friction of spur gear teeth is mainly dependent upon the 
form of the tooth outline, and may be investigated ,by consider- 
ing the form, extent and position of the line of action. In most 
cases the friction is proportional to the duration of action e. A 
coefficient, dependent upon the position of the line of action 
may be determined from f, and may be taken = j-i, when the 
arc of action is equally divided on both sides of the central po- 
sition ; as in the case of epicycloidal teeth ; and = i, when, as 
in many cases, such as pin tooth gearing, the arc of action is 
entirely on one side of the centre ; while for evolute teeth it 



may be taken : 



3/ 



that being about midway between the two 



preceding forms. The tooth friction is also greatly dependent 
upon the number of teeth in both wheels, being proportional to 
their harmonic mean, and it diminishes rapidly as the number 
of teeth is increased. 

If we make the coefficient of friction =yand take the num- 
ber .of teeth as Z, and Z^, we have for the percentage of loss pr 
in tooth friction : 



a. Epicycloidal Teeth 
I I 



1 



/ 



r/ 



Z~ Z^ 

b. Evolute Teeth. 



Pr 






c. Pin Teeth.* 



Pr = -.f 



■ • (>90 



The value of the coefficient of friction/" is in no case small, 
even when the teeth are well lubricated, on account of the usual 
high pressures ; a usual value maj- be taken, /^ 0.15, while for 
new and dry wheels it reaches 0.20 to 0.25 and even higher. 

The minus sign in the formula is to be used when one of the 
wheels (Z^) is an internal gear. 

ExajiipU I. In a pair of epic^-cloidal gears, of seven teeth, the value of e = 
1.225. TakingX= 0.15 we have according to (191 a) for the loss by tooth fric- 
tion; 

2 1.225 
^— 3.14X0.15 X — X =0.0824, or about 8J^ per cent. 

Example 2. Epicyctaidiil Teeth. ^^^^^40,6 = 1.44 andweget: 

= 0.0169, or about 1.7 per cent. 

Example 2,. Epicycloidal Teeth. .2=7, Z\ = — 60 (internal gear). £ = 1.40 
and we get : 



A = 3.14 X 0.15 X — X ■ 
40 



/r=3 14X0.15 (J — sV) - 
Example 4. Epicycloidal Teeth. Z= 



-^ — = 4.2 percent. 
1, Z\ = K (rack), e = 1.37 
4.6 per cent. 



1-37 



Example s. Pin-tooth Gearing. Z^6, Zj = 40. We have, as determined 
by construction, as in Fig. 5S3, e = 1.166. Hence we get from (191 c) : 



A=3.i4Xo.ls(i+^5)Xi.66 
Example &. Evolute Teeth. ^=^1 = 40. e 
3X 1.92 



■ 2.6 percent. 

= 1.92. We have from (igi <5): 



/>r = 3-14X0.15 X — X 
40 



= 3.4 per cent., or double that in Examp. 2. 



It will be seen that the tooth friction is least with epicycloidal 
teeth and greatest for pin gearing ; evolute teeth being midway 
between. 

The wear upon gear teeth is aifected by other considerations 
besides that of the coefficient of friction, the pressure of the 
teeth upon each other, and the relative rubbing movement of 
various portions of the profile also entering into the problem. 
The wear is therefore not constant for a constant pressure, and 
it is an error to assume, as is sometimes done, that the form of 
evolute teeth is unaltered by wear. These teeth usually show 
the greatest proportional alteration by wear, since the flank of 
the tooth below the pitch circle has a very much less rubbing 
movement than the portion of the opposing tooth which rubs 
against it and hence the wear is unequal. 



^ Approximately. 



THE CONSTRUCTOR. 



135 



The effect of this may frequentl3' be observed in practice, 
where the smaller of a pair oi evolute gear wheels -will be no- 
ticed to be worn into deep hollows below the pitch circle. 

The conclusions given above about the percentage of loss may 
also be determined geometrically iu the following manner : 

Take the two portions of the tooth profiles ivhicli wor/i together 
and divide each by the chord of the corresponding portion of tlie 
line of action, multiply each result by the ratio of the length of 
its poiiion of the line of action to the entire length of the line 
of action, and then multiply the sum of the two quotients by the 
coefficient of friction. 

The result will be the percentage of loss, pr. The chord re- 
ferred to becomes the line of action itself in the case of evolute 
teeth. This method serves also for pin teeth, and is verj' useful 
for the designer, as the data can all be taken off the drawing 
with the dividers. 

§ 214. 
Generai, Remarks on the Foregoing Methods. 

Each of the preceding methods possesses its merits and dis- 
advantages. 

Epicycloidal Teeth. These possess the great advantage that 
they -will work together iu any series with as few as seven teeth, 
while for evolute teeth the lowest in series is 14 teeth, and in 
no case fewer than 1 1. The loss from tooth friction is a mini- 
mum with this form, and the wear less injurious to the shape of 
the tooth. The minor objections which have been raised are 
that the double curve increases the diiHculty of construction, 
and that any variation of the distance between centre causes im- 
perfect action to follow. 

Evolute Teeth. The advantages of this form are that the 
simple shape is readily made and that any variation of the dis- 
tance between centres does not affect the action. 

Against these must be set the fact that for low numbered 
pinions the flanks must be altered to avoid interference, or the 
tops of the teeth must be taken off. The fact that the distance 
between centres may vary is rather an objection iu many cases, 
as the arc of action is reduced, and iu transmission of heavy 
power the shocks upon the teeth are liable to be increased. 

Evolute teeth are well suited for interchangeable gears, if low 
numbered pinions are not required (30 teeth being the minimum), 
and where but small power is to be transmitted they are excel- 
lently adapted. For wheels which run only in pairs, and hence 
for bevel gears, this form is excellent, since it is so readily made. 
See \ 222. 

Pin tooth gearing and the mixed outlines are only used for 
special work, such as in hoisting machinery and the like, and 
in such cases the wheels are often made of wrought iron or steel. 

Disc wheels have a very limited application, but iu some spe- 
cial forms of mechanism they are very useful, and will be dis- 
cussed further. See Chapter XVIII. 



If bevel gears are required to interchange (see \ 200) they" 
must not only be of the same pitch, but nmstalso have the same 
length of contact line, A S, Fig. 596. Since these conditions 
are very infrequent, it follows that bevel gears are generally 
only made to work in pairs. In practice it is found that a vari- 
ation of less than 5 per cent, in the length of *he contact line 
may be neglected. Gears of the same pitch and same angle of 




Fig. 596. 



axes, but with a small variation of contact line, are called 
"bastard gears.'' A pair of right angled bevel gears of 80 and 
45 teeth, might be altered in practice, if required, into bastard 
gears of So (i ±0.05), /. c., 84 to 76 teeth, which would work 
with the other gear of 45 teeth. 

I 216. 

Construction Circles for Bevei. Gears. 

The geometrical figures which are formed by one cone rolling 
upon another, require that both cones should have a common 
apex. The surface thus developed is called a spherical cycloid. 
Of these there are five particular forms, as with the plane cy- 
cloids, the latter being really those for a cone with an apex: 
angle of iSo°. The spherical cycloid is very similar in form to 
the plane cj-cloid, as are also the corresponding evolutes ; the 
branches of the curves assuming a zig-zag form.* 



B. CONICAL GEAR WHEELS. 
?2IS. 

Generai< Considerations. 

In the case of conical gear wheels, or as they are generally 
termed. Bevel Gears, the working circles of a pair of gears which 
run together, lie on the surfaces of a pair of cones, the apex of 
each cone being at the intersection of the axes of rotation. In 
such case the pitch circles are taken at the/> base circles of the 
respective cones, as S D, and S E, Fig. 596. The length of the 
teeth is measured on the supplementary cone, to each base cone, 
SB being the supplementary cone for S D, and 6' C that for 5' 
E, B C being at right angles to A S. The length of teeth is laid 
off on SB and SC, and the width of face on SA; the tooth 
thickness being spaced off on the pitch circle and all the teeth 
converging to the point A. 

The respective radii S D and S E of the two cones are found 
by dividing the angle a of the axes, in such a manner that the 
perpendiculars S D and S E let fall from Sto the axes, bear the 
same ratio to each other as do the numbers of teeth, or inverse- 
ly as the number of revolutions : thus S D : S E = Z : Z^ = 
«, : n. There are, therefore, two solutions possible, according 
as the pitch line S A is taken within the angle o, or in its sup- 
plement ; or what is the same thing, according to which angle 
is taken as the angle of the axes. The difference between the 
two consists iu the fact that for a constant direction of revolu- 
tion of the driving shaft the driven gear revolves in one direc- 
tion for the first solution and in the opposite direction for the 
second solution. One of the solutions gives an internal gear, 
when «i : n <^ cos a. 




Fig. 597. 

The use of the spherical cycloid for the formation of bevel gear 
teeth would involve many diificulties. In order to construct 
such teeth, it is therefore common to use the method (first de- 
vised by Tredgold) of auxiliary circles, based upon the supple- 
mentary cones, and enabling the teeth to be laid out iu a simi- 
lar manner to those of spur gears. The auxiliary circles for the 
bevel gears R and 7?,, Fig. 597, are those of the spur gears hav- 
ing the same pitch, their radii being respectively r and }\, the 
elements B S and CS of the supplementary cones. 

For any given angle a between the axes, the radius r, and 
number of teeth 3, for the auxiliary circle can be determined 



*See Berliner Verhandlung 
the Spherical Cycloid. 



1S76, pp. 321, 449, Reuleaux, Development of 



136 



THE CONSTRUCTOR. 



from the radii R and A",, and tooth numbers Z a.niL Z^, by the 
following formula : 

r _ y/R'- -^ R^- -Y 2 R R^ cos g 
R "^T^ cos~a; 



z^ _ VZ' + Z^ + 2 ZZ^cosfl 
^ Z^-\- Z cos a 

If the axes are at right angles, we have 

r \/R' + R'' z \^Z-+Z;'^ 



R 



r^m 



Z 



Zy 



(192) 



Example. — A pair of bevel gears have 30 and 50 teeth, and an angle between 
axes a. = 60°, hence cos a = J^, and we have for the auxiliary circle of the 30 

. /.T^^ 

tooth gear : . 



V. 



V^3°^ + 50^ + 2 . 30 . 50 . OS 

50 + 30.0.5 ~" 

\/ 4900 



For the 50 tooth gear we have also ; ^i = 50 



■ =32-3> say 32. 



= 64. 



30 + 50 . 0.5 

From these numbers and the given pitch, the auxiliary circles 
can be laid off and the teeth drawn. 

Low tooth numbers are not available for bevel gears, since the 
errors which are involved in the method of auxiliar}' circles be- 
come disproportionately great. By using not fewer than 24 
teeth for the bevel gear, a minimum of 28 for the auxiliary cir- 
cle is obtained, and the evolute system can be used to advant- 
age. This form of tooth is best adapted for this purpose, on 
account of its simplicity of form, notwithstanding the minor 
defects which have already been noticed. 

The loss from tooth friction in bevel gears is approximately 
equal to that of their corresponding auxiliary gears. 




Fig. 598. 

l 217. 

The PL.4NE GE.A.R Wheel. 

Internally toothed bevel gears are not used, on account of the 
practical difiiculties involved in their construction. There is, 
however, an interesting form of gear wheel which lies interme- 
diate between the external and internal forms. If the numeri- 
cal ratio between a pair of bevel gears is = cos a, one of the so- 
lutions for the base cone gives for the latter a plane surface, 5 
E, Fig. 598. 




Fig. 599. 

The supplementary cone in this case becomes a C3dinder, and 
the radius of the construction circle becomes infinitely great, 
hence the tooth outlines are similar to those used for rack teeth. 
If the evolute system is used the teeth are very simple, and the 
plane gear in some cases becomes a very convenient form of 
construction . 

As already stated, the ratio is 

— ' = cosa (193) 



from which, if for example a ^ 60°, we have -~ = ^. If the 

angular relation of the axes is given it follows that but one ve- 
locity ratio can be obtained. This is determined from the angle 
72, which is one-half the apex angle of the cone R2, and from 

the ratio ---? = sin y^. 

It is sometimes very convenient to arrange a plane gear so 
that it may work with both of a pair of bevel wheels. This is 
shown in Fig. 599, in which the gears j?,, R^ have the semi-apex 
angles y.^, J's, and have their axes at right angles. We then have : 

R, , 

~ = tan y,, = cot 73, 

R3 

from which we obtain the following values : 

-^ = tan 3-2 = 1 i i f I I 2 3 4 



72^ 

J?, 



= Sm }'., : 



14° i8°3o' 26°4o' 36°5o' 45° S3°io' 63°2o'7i°30 76° 
= 0.2420.317 0.449 0.6000.7070.8000.8940.9480.970 



Either of the wheels 7?,, .^3, can be used with the plane gear 
Ri if the number of teeth have the ratio given by the value of 
sin 7.,. Although this limits its application, yet the plane gear 
is frequently found ver}' useful for angular transmissions.* 



C. HYPERBOLOIDAL GEAR WHEELS. 
I 21S. 

B.\SE Figures for Hyperboeoidae Wheees. 

Hyperboloidal wheels are used to transmit motion between 
inclined, non-intersecting axes. The figures upon which they 
are based are hyperboloids of revolution having a common 
generatrix. These may be determined in the following manner. 




Fig. 600. 

In Fig. 600 is given a projection normal to the line of shortest 
distance between the two a.xes. The angle a is divided into two 
parts /3 and ft, in such a manner that the perpendiculars let fall 
from any point A, of the line S A, upon the two axes, shall be 
inversely proportional to the revolutions of the gears. S A \% 
then the contact line of the hyperboloids \ A B = R' and A C 



*The so called "Universal Gears" of Prof. Beylich, introduced in 1866, 
should'be considered as a variety of conical gears in which the angle of the 
axes may be conveniently varied. These may be used for axes of angles 










varying from 0° to 180° As shown in the illustration, these wheels are 
formed of globoids of the III Class (see g 224), the meridians ibrming the 
teeth and spaces. They have found but limited application. A model of 
these gears is in the kinematic cabinet of the R03'al Technical High School. 



THE CONSTRUCTOR. 



137 



= ^'1, are projections of the radii of the hyperboloids intersect- 
ing at A. We have 

(194) 

The actual radii R and R-^ are yet to be determined, as well as 
the radii S D =: r, and S £ = r^ of the gorge circles. 
For the latter we have : 



also : 



tan/3 



tan ft 



(195) 



1" cos a 



that is, r and i\ have the same relation to each other as the por- 
tions A F a.nA. A G of a perpendicular to the line of contact. 
If we call the shortest perpendicular distance between the axes 
= a, we have ; 



1 + 



-Hi)" 



a 



I -| cos a 

n 



1 + 2 



—^ cos « + I — i 1 

n \" ) } 



(196) 



The radii R and R^ are hypotenuses for the triangles whose 
sides are R' and ?', R^' and 7\ (see the left of the figure) or : 



R, = VR'^- + r{- 



(197) 



R' and R/ being determined as above, when the distance S A 
= / is given. For the angles /3 and /?, we have the general ex- 
pressions : 

sin a 



tan/3: 



tan /3i = 



4- cos a 



"1 1 

— - -{- cos a 



(198) 



As in the case of bevel gears, two solutions are possible ac- 
cording as the angle a, or its supplement, is taken in determin- 
ing the line of contact S A, Fig. 601. The choice of solution 




Fig. 601. 



governs the direction of rotation of the driven gear, and one of 
the solutions renders it practicable to make an internal gear ; 
although this construction has been little used, and has but little 
practical value. 

If the angle of the axes a = 90" we have 



= tan^ /3 = 



(?)" 



(199) 



a 



tan/3 = 



n^ -( 

«^ -I- «i^ ' 

n"" -\- 11^ ' 
"1 



(200) 



In the construction of the wheels, corresponding zones are 
chosen on the two hyperboloids. If the distance between the 
axes is small, the zones Ij'ing in the gorge circles are generally 
unsuitable, but when the distance is greater they may be used 
and the figures approximated by truncated cones. 




Example I. a = 40°, 



Fig. 602. 

= Vsi (see Exaraple i, in g 



221), a = 4". 



Also tan 6 = 



f 0.5 + COS 40° _ 

ri 2 + cos 40° ~ 

g _ 1+2 co s 40° 

r 1+2X2 cos 40" 

r = 0.31398 X 4= 1-256", 
ri = 4 — 1.256 = 2.744". 
sin 40*^ 0.642S 



2.766 



8.064 



= 0-31398, 



: 0.232393 = tan 13° 5', and 



2 -f cos 40° 2.766 

f 1 = 40° - /3 = 26° 55'. 
Ifwetake5'y« = /= 8" we have .A" = /sin 13° 5' = 3 X 0.126368 = 1.81" 
J?l'= 8 sin 26° 55' = 8 X 0.452634 = 3.62"; finally 

:" and 



— - — — or say tlie number of teeth Z = ^6, and 



.S' = v/ (i.Si)= + (1.256): 
R\ = \/ (3-62)= -I- (2.744)= = 4-54". 
Exa7nple 2. a ^ 90' 
^1 = 20 ; a = 0.75". We have from (197) 

r / g \- Si 

— = — = = 3.24 and from (200) 

ri \ 5 / 25 ^ ^' 

^ aX 9 - ^ o 75 X Si 

S- + 9- 106 

0.573 



= 0.573" 



and j'l = 



For ^, we have tan ^ = ■ — - 



3-24 
■ i.S, hence (S = 



■ 60° 57', and ft = 29° 3'. 



If we make 2? = 2' 



, we have from (197); 



7;" = .^ ^- . — r- = \/-2- — 0.5732 = 1.916'', 

and hence .^i', according" to (194) is = g J?i = 1.063", hence 

Jii = \/ 1.0632 + 0.1772 = 1.07S". 

The appearance of such a pair of gears is shown in Fig-. 602. According 

to the table in ? 202 the pitch for the larger gear is : / ^ = = o.-sk", 

5-73 5-73 

and for the smaller gear fi = -^ — — = 0.339". 
3.18 

Example 3. a = go°, -^ = 1, ^ = 45°, r — r\, R — R^. In this case the hy- 
perboloids become similar (see Example 4, \ 221.) 

-Exainple 4. In the special case in which — = cos a, and the position of 

the contact Itlie, which is determined byiS, lies in the supplement to a, 'so 

that — = cos a, the base figures become, the one a normal cone and the 

other a plane hyperboloid, see Fig 603. This construction is similar to the 
preceding forms of plane and bevel gears, and may be conveniently used to 
work with a train of common bevel gears, although but few practical appli- 
cations occur, partially owing to the fact that the prolonged axis of the bevel 



I3S 



THE CONSTRUCTOR. 



gear passes through the plane gear. For a — 60" 



obtain the plane gear. We have tan (3 = 

R' sin 30° 

Also -jr-T- = — :— — ^ = 0.5 ; r = o,r\=a,R 



i \/3, f- 

= R', R\ 



-i = — Vz = — cos 60^^ we 
n 

= 30°, tan ^1 = 00 , ^1 = 90°. 



If — — be negative and less than cos a. we obtain an hyperboloidal internal 

71 
gear. 




Fig. 603. 

Rack teeth may also be constructed to work with hyperbo- 
loidal gears. In this case the teeth of the rack are inclined 
while the pinion becomes an ordinary cylindrical spur gear, 
since in order to satisfy equation (195) with »'; = 00, the angle 
/? = (7, and /3i = a, see Fig. 604. Applications of this construc- 
tion may be found in various machine tools. 




Fig. 604. 

\ 219. 

Teeth for Hyperboi,oidai, Gears. 

The construction of the exact forms for the teeth of hyper- 
boloidal gears is a very difficult operation, and in practice an 
approximation is used similar to that employed for bevel gears. 
The method adopted is to determine the supplementary cone to 
the hyperboloid used, and as in the case of bevel gears, use the 
corresponding construction circle. 




I ■ 

Fig. 605. 

The apex // (Fig. 605) is determined by drawing ^//"per- 
pendicular to the generatrix 5 A, which, as before, is taken 
parallel to the plane of the drawing. The teeth will be formed 
with sufficient accuracy' if two construction hyperboloids are 
taken with the same angle of contact as the base hyperboloids, 
according to the conditions in (19S) and (199), and the teeth are 
formed on the surfaces, which are described by the edges of the 
construction hyperboloids upon the base hyperboloids.* 



If it is desired to approximate to the hyperboloidal zone by 
the use of a conical surface, the apex must be determined. In 
this case the generatrix S A \s rotated about the axis H S until 
A falls on the pointy of the circumference, when the new pro- 
jection of the generatrix will pass through the apex M of the 
cone. 

The tooth friction of hyperboloidal gears is necessarily great. 
This will be considered later, in connection with the speed of 
the rubbing surfaces, which is similar to that of the spiral gears 
which are tangent at the gorge circles (see ^ 220.) 



D. SPIRAL GEARS. 



Cyi,indricai< Spirai< Gears. 



Cylindrical spiral gears may be used in the same m?nner as 
hyperboloidal gears for the transmission of motion between in- 
clined axes, and in some.cases possess advantages over the lat- 
ter. There are a number of useful variations of spiral gears. 




Fig. 606. 

In Fig. 6o5 is shown a pair of wheels, A and B, both with left 
hand spirals and corresponding tooth profiles. The pitch angles 
y and ;'i are so chosen that at the point of contact the pitch cy- 
linders have a common tangent, so that if a be the angle, of in- 
clination of the axes, y + jj -|- a ^ 180°. If we indicate by w 
and v^ the circumferential velocity in the direction of the tan- 
gent and normal respectively, we have : 

v^ sin y , iu R sin y Z , . 

— !- = — = whence — -^ i — =: — . . . (201) 

V sin 7j !i J?i sin 7; Z^ 

The normal pitches, J = / sin 7, and ij = /^ sin 7i must be equal 

to each other, whence — ^- ?. 

fi sm y 

As indicated by the components of velocity v' and v/, there 
is an end long sliding action of the teeth upon each other, with 
a velocity : 

c'= z/-}- i// = <; (cot 7+ cot 7i) (202) 

This sliding consumes power and causes wear, and will be at 
a minimum when z.' and z';' aje equally great, that is when 

7 = 71- 

With regard to the choice of y and 7, the conditions may be 
so taken that the position of the coinciding tangents of the two 
spirals shall be slightly before or slightly after the actual line of 
contact, but as close as may be possible. This is similar to the 
position of the line of contact of hyperboloidal gears (g 21S) 
and may be stated as follows : 

"' 
R cot 7 

R-^ cot 7i 



■ -\- cos a 



(203) 



- - -|- cos a 



as also 



cot y =r. 



(204) 



* See Herrmann's Weisbach's Mechanics, II. ed., Ill, i, p. 418 ei seq. 



--{• cos a 



THE CONSTRUCTOR. 



139 



For a = 90° we have cot y =: — 1-. Such spiral wheels, when 

11 
the teeth are well made, transmit motion very smoothly, but the 
surface of working contact is very small. When the axes are 
at right angles and the wheels the same size, it is often incon- 
venient to use spiral gears on account of the large size required. 




ExafiiJ^le. Fig. 607. Let - 



Fig. 607. 



: 3, and a = 90°. 'vVe have from (203) 



R 



m^ 



■ 9 and from (204) cot 7 = = 3, whence y =^ 18° 26' and vi ^ 71'^ 

n 

34'. The sliding velocity is c' = c (3 + 0.333) = 3j ^- The small value of the 
angle y makes it undesirable to use the smaller gear as the driver. These 

objectionable features are of increasing importance and for example, — ^ 

:= 5, and ■ = 10, we get — rj— = 25, and 100, and y about iij° and 53°. The 

difficulty of cutting the teeth on the lathe also increases, as may readily be 
seen. 

? 22r. 
Approximately Cylindrical Spiral Gears. 

If, of the preceding conditions, only those of formula (201) 
and (203) are strictly observed, the difSculties of construction 
are much reduced and at the same time satisfactory wheels ob- 
tained. 

Three methods may be employed : (a) a slight modification 
from the correct spiral form may be given to both wheels, (d) 
one gear may be made a true spiral, and the variation all thrown 




Fig. 608. 




In many cases the worm is made a true spiral and the conse- 
quent wear disregarded, but in more careful work the method 
(b) is adopted and the worm wheel cut with a hob, which makes 
the proper modification in the shape of the teeth. 

The friction between the worm and teeth of the worm wheel 
is very great, as the thread slides entirely across the teeth. We 
have for the coefficient of fraction y, for the ratio between the 
actual force /" and a force P acting at the same lever arm on 
the screw, but free from frictional resistance, approximately : 



~P'' 



I +/. 2Tr J? 



ft 



ZTT R 



Fory = o.!6 we have practically 

P' , R 



It follows that to obtain the minimum of frictional loss. 



(205) 
R 



must be made as small as practicable. 

Morin gives the rule R = ;^ t, which makes 

P' 



P^ 
~P 



4; Red- 



tenbacher makes R = 1.6 /, whence — - = 2.6. If we make 



P' 
7? = i', we get — ■■ 



2, and this is as low as 



R_ 
~T 



can well be 



made. In this case it will be seen that a higher efficiency than 
50 per cent, cannot be obtained, and it is also ajaparent that the 
worm must be the driver, since the resistance of friction would 
just balance the reverse driving action. The ordinary tooth 
friction and the journal friction must of course be added. 




Fig. 610. 



Fig. 611. 



Fig. 612. 



The tooth outlines for both worm and wheel are the same as 
for a rack and gear wheel, taken on a longitudinal section 
through the axis of the worm. The evolute tooth is especially 
applicable, and Z-^ must not be less than 2S (2 209). The surface 
of contact is theoretically only a mathematical point, but in 
practice there is a small flattened surface of contact, and if a 
larger surface is desired the wheel must be cut with a hob of the 
same form as the worm which is to work with it. 

Wheels which have a contact bearing of a point only, may be 
called precision-gears, as distinguished from power-transmitting 
gears. The difference, however, cannot be sharply maintained, 
for as already shown, worm gearing is used for the transmission 
of both large and small forces. 

The possible variations of the pitch angle permit a great va- 
riety of spiral gear combinations, as the following examples 
show : 



into the other gear, or (t) the wear which is at first caused by 
running the approximate forms together may be disregarded 
until the parts have worn themselves iuto smooth action. From 
these reasons a widely varying practice in the construction of 
spiral gears will be found. One of the most important applica 
tions is that of the worm and worm wheel, Fig. 60S. In this 
case a = 90° and .2' = i , the teeth of the wheel R^ being inclined 

at an angle 7, with the edge of the wheel, whence tan y = - 

In the arrangement shown in Fig. 609, we have 



= 0.15916 



R 



o ^ 90 — y and the teeth on R^ are made parallel to the axis. 

The pitch of the screw is here made = — ! — 

cos 7 

■wheel. The velocity ratio of transmission, according to the 
fundamental formula (186) is it^:n ^ Z : Z^, or this case it 

equals — -.* 



Example I. Given — = i, the perpendicular distance between axes a = 

R -\- Ri, and the angle between axes a = 40^. If we make y = 60°, we have 

from (3 220)71=1 So — 40 — 60 = 80° (see Fig. 610), and from (201) -5- — ^ - 

sin 80° 0,5 X o. 



^' sin 69° 0.S660 

determined. Ifwemake«== 



Sill y n 

= 0.56S6, from which J? and .^i may be readily 
l" we have 



-ft 



1.5686 



2.55 



and R = i»45"- For Z= 20, Z\ = 40, the normal pitch r = ^ sin 7 - 
2 X T X 1-45 X 0.866 



■ = 0.272 X 1-45 = 0-394 ■ 



for a pitch t^ of the xhe circumferential pitch t = -^^ 



0.394 ^ 



sin 7 



■ °.454", h = 



°.394 



tR sin-) 



: 0.400'. 



o 9S4S 
c (cot 60° + cot 80°) = c (0.5774 



The sliding velocity c\ according to (202) 
-1-0.1763) =0.7537. ... 

Example 2. In order to make c' a minimum, we may make 7 = 71 



iSo — a 



= 70'- 



, see Fig. 611. 



<..t(,(,. 



2 X IT X 1-333 X 0.9397 



* In the illustration Z-^ = 30, which in (203/ for a true spiral would requi 
.ffl — gocff, and y = 88.1°. 



We then have -=- = J, i?i 
.'n 
o. 394 

= 0.394", /= i*! = = 0.419, ana 

20 " 0.9397 

c' = 2 cot 70° X <; = 0.728 c. It will be seen that the value of c' in Example 1 
approached very closely to the miuimum. 



I40 



THE CONSTRUCTOR. 



Exmttpls 3. If so desired we may make 7 ^ 90°, when one wheel will be- 
come an ordinary spur gear. Fig 612, and we have 71 = 180 — 40 — 90 = 50°. 

-— - = 0.5 X 0.7660 — 0.383, iVi = 2. So", R = I. II'', T = 0.348 ' , t ^ r, /i = 0.454", 

C" = 0.8391 c. 

If instead of a, the normal pitch t is given, as is generally the 
case with hobbed -worm wheels, we choose ;- and j-j and then 

have Ji sin 7 = '—, whence : 

2 IT 



2 5r sin ) 



R,- 



Z,: 



(206) 



2 IT sm y^ 
Both R and r may be given, when }■ must be determined, and 



we have ; 



sm }•=■ — 




Fig. 613. 




(207) 




Fig. 6t5. 



The following examples illustrate a variety of cases : 



Example 4. 
180 — 90 



45" 



: 90°, Z ^ Zy The sliding to be a minimum, hence v — Vi 
The two wheels are similar, both being left hand or as in 



Fig. 613, both right hand. The sliding velocity is c' = 2 col 45^ X c = 2 c. 

Exaniple s- In the arrangement shown in Fig. 614 there is added to the 
right angled pair A B. a third wheel C, also right angled, when the wheels 
^ and C will revolve in opposite directions. The middle gear B reverses the 
motion, as in the case of bevel gears. 

Example 6. When a = o, the axes are parallel and a pair of spur gears 
with spiral teeth is obtained, this form being called Hooke's or White's 
gearing, Fig. 615. y and y^ together include 180°, and one gear is left, and 
the other right hand. In this case the teeth are formed in true spirals. In 
this case the sliding velocity c' — o. For the wear on this form of gear see 
1 222. When (x = o andy=s o the wheels become spur gears. 



Fig. 617. 






Py-i- • • 



Fig. 61S. 



Fig. 619. 



If the other limit of spiral gears is reached some noteworthy 
forms are obtained. 

Example 7. a = 90°, 7 = 10°. 71 = 80°, i?i = w. This gives a rack and screw, 
Fig. 616. If 7j ^ go° and the teeth normal, 7 = 10*^ and a = So°, and the teeth 
of the rack correspond to the section of a nut. In the Sellers planing ma- 
chine the rack teeth are placed at sucli an angle that the lateral pressure 
just balances the opposing tooth friction. 

Example 8. j? = 7?, = co. This gives two racks, sliding in each other, Fig. 
618. We have, as before, z'l : v = sin 7 : sin yj. If a = 90°, as in Fig. 619, and 
7 = 71=45°, we have v^=vi. This construction is used in some forms of 
boring machinery for cannon, and in screw cutting machines. 



Example 9. a = 90°, yi= 90°, also y = o, both radii of indefinite magnitude. 
Fig. 620. This is the so-called revolving rack, used on governors and similar 
apparatus in which endlong motion is to be transmitted from a revolving 
piece. The velocity ratio of .-1 to ^ = o. 

Example 10. The worm, or endless screw, as already stated, is a form of 
spiral gear wheel. These are two special forms of worm gear which although 
seldom used, are of interest. 1 here are the forms of internal gearing shown 
in Figs. 621 and 622. In the former the worm wheel is the internal gear, 
while the latter shows an internal worm, with external or spur worm-wheel. 




Fig. 620. 



Fig. 621. 



Fig. 622. 



I 222. 

Spiral Ge.4r Teeth and their Friction. 

Spiral gears are cut in a similar manner to screws, the tool 
being carried in the slide rest of an engine lathe, and set at the 
proper angle. The pitch of the screw thread is : 

s ^=2- R tan 7, 

and the travel of the rest is effected \>y proper change gears, 
according to the selected values of 7 and )'j.* 

The tooth outline to be used is determined according to the 
radius of curvature of the supplementary spiral, that is, to that 
at right angles to the spiral to be cut. The radii of curvature 
r and r^ to be used are : 

R R\ I o\ 

r=—.^ — ,7'. = ^-J — (208) 

sin - 7 sin - 7i 

These give the radii for the construction circles to be used 
with the pitch r ; the shape of the tool with which the teeth are 
cut is then determined. 

Exajnple I. For the wheels of the first example in the preceding section , 
we have : 

— '-15 _ » , ^ 2-55 
''~sin=6oO -'^ ' ' sin = So°' 

If it is preferred to determine ?', graphically from formula 
(20S) the method given in \ 29 maybe employed. 

The frictional resistance of spiral gearing is often a matter of 
much importance. If the frictional resistance is assumed to be 
zero, we have for the relation of the force /"applied to the driv- 
ing wheel, to the force Q delivered by the driven wheel : 



2.58". 



P 



sm 7 



Q 



sm 



(209) 



The ordinary tooth friction, which is the same as that of the 
construction gears (see J 213) to which must be added the fric- 
tion due to the sliding of the teeth, whenever a. is greater than 
zero. The value of the latter friction is governed by the sliding 
velocity c' . For the calculation of the loss of useful effect we 
may use the formula : 

P' sin -/i sin (/ -I- ^) , 

P"'~sinT' ^hItT— <) 

in which 1^ is the angle of friction for the coeiBcienty] whence 
tan =y. Fory"= 0.16 we have (6 = 9°. 

Example 2. For the wheels in the preceding example we have 
P' _ sin 80° sin 69° _ 0.9S48 X 0.9336 _ 
P sin 60° sin 71*-" 0.S660 X 0.9455 

To this must be added the ordinary friction of the equivalent 
spur gears. Another source of loss is that due to the lateral 
forces K and K-^, acting in the direction of the axes. For these 
we have 

-^, = cot(7+rt, ^' = cot()-,-iJ). . . . (211) 

Example z- For the preceding- .erears we have A'= P' cot 6g° =0.3839 /^^ 
Kx = Q cot 71'^ —0.3443 Q from which values, in connection with the known 
dimensions of thejounials the corresponding resistances can be determined. 

When a =£?, that is, for parallel axes, the sliding action of the 
teeth is zero, and the value of P^ in v-'^) is the same as P\ 
hence spiral gears for paraiiel axes work without the tooth fric- 
tion due to lateral sliding, the ordinary tooth friction alone re- 
maining, as well as the forces K and A'j. 



* Brocot's Tables will be found of service in arranging change gears, 
(Calcul des Rouages par Approximation. Paris, 1862). 



777.5' CONSTRUCTOR. 



141 



The tooth friction may be reduced to a very small amount by 
reducing the beariug surface of the teeth of one gear to a point 
of contact, or practically to a knife edge. Such gears (devised 
by Hooke) are only of use for purposes of precision, but in some 
cases are found serviceable.* 




Fig. 623. 



Fig. 624. 



Instead of the edge bearing, a rounded surface may be used, 
with its highest part corresponding to the lineal bearing as al- 
ready shown by Hooke and by Willis. The tooth outlines for 
both gears are determined as usual, and then one or both pro- 
files are redrawn within the original curves, Fig. 623, and the 
modified outline.'^ used to form the tooth spiral ; teeth so con- 
structed running nearly free from friction. In such cases the 
length of flanky, and face k may be reduced as shown. Such 
forms are more properly to be considered as screw thread pro- 
files than as gear teeth. Willis has shown that in both gears 
the flanks may be made radial and the crown of the teeth semi- 
circular, Fig. 624. Since such teeth are weakest at the base, it 
is preferable to use a modified form of the evolute tooth. Fig. 
625. This may be approximated to by using a circular arc of 
smaller radius than B S =: R cos a, the centre B' being taken 
on the normal N N, through the point of contact. 



B..- 

H..,,--v 





Fig. 625. 



A similar form to the preceding gears is the so-called step- 
gearing, Fig. 626, frequently used in planing machines (by 
Shanks, Collier and others). The tooth profiles may be modi- 
fied as above, to reduce friction, but the gradation .j should be 
as great or greater than the pitch t. Fewer than four sections 
should not be used. 

An objection to the use of spiral gears is the axial pressure 
K, this, however, can be eliminated by the use of double gears 
of opposite inclination. Such gears have been known for a 
long time (White, 1808) and for moderate service, have been 
frequently used, as in spinning machinery, tower clocks, etc., 
and more recently they have been applied to heavy work, nota- 
bly for rolling mill gearing, both in Germany and America. 

The pinions used in rolling mill work are made with 9 to 16 
teeth, with pitch diameters from 4" to 24" and over. Evolute 
teeth are used, with a base angle from 62 to 69°. The face length 
of the teeth is made about 0.22 t. 

If the evolute curve is accurately made, the tooth contact is 
practically the same as with ordinary spur gears, and the surfa- 
ces of contact can readily be discerned, extending diagonally 
across the teeth. When such a surface of wear is visible, of 
course the teeth are not free from friction. Fig. 627 shows a 
cast steel pinion of ten teeth, for rolling mill service. This gear 
is cast in one piece with its shaft and coupling ends, although in 
many cases the shaft is made separately. 



The space .? between teeth at the middle of the gear, is called 
in the Westphalian shops the "spring'' ,of the teeth. If it is 
desired to approximate to the frictionless action of the teeth, 
this " spring " must be slightly greater than the pitch. 




Fig. 627. 

For very large transmissions the gears may be made in two 
parts. Fig. 628 shows a pair of such gears for a reversing roll- 
ing mill by the Hagener Steel Works. The pitch diameter is 
43-3", the pitch S|<", the face of each gear 20", and the total 
weight 24,200 pounds. The teeth are made with double reverse 
angles on each gear, so that the conditions are the same when 
running in either direction, and the whole is a masterpiece of 
machine work in steel. 




Fig. 628. 



223 



Spirai< Beveiv Gears. 

Spirally formed teeth are sometimes used on bevel gears, and 
in this case the distance a, between the axes becomes zero, while 
the angle a remains to be given. For the curvature of the teeth 
it is best to use a conical spiral of constant pitch, the projection 
of which on the base of the cone is an Archimedean spiral. 
Frequent applications of such wheels are to be found in spinning 
machinery, and they are operated successfully at quite high ve- 
locities."^ 




Fig. 629. 

The same varieties may be made in bevel, as in spur gears, 
and in Fig. 629 is shown a reverse spiral bevel gear of cast iron, 
as made by Jackson & Co., at Manchester. Similar gears are 
made of cast steel by Asthover 6c Co., at Annen in Westphalia. 
Stepped teeth are also used in bevel gears, and in Fig. 630 is 
shown such a wheel by A. Piat fils, of Paris. 



* These gears have been used in physical apparatus by Breguet for speeds 
exceeding 2000, or according to Haton, as high as Sooo revolutions per second 
or 480,000 per minute. 



* For a machine for the correct construction of the teeth of spiral bevel 
gears, see Genie ludustriel. Vol. XII. p. 255. 



142 



THE CONSTRUCTOR. 




Fig. 630, 

I 224. 

Globoid Spiral Gears. 

If a circle is revolved about an axis A A^ coinciding with one 
of its diameters, and at the same time a radius C^" is moved 
about the centre C, with an angular velocitj' proportional to that 
of the circle itself, the circle will generate a sphere and the 
point of the radius which is at the surface of the sphere will 
trace a form of spiral curve. This may be called a spherical 
spiral,* and adjoining lines of the spiral on the same meridian 
are equidistant. 





Fig. 632. 

If the radius C 5" passes the axis of rotation, the new spiral 
will intersect the one previously traced, as at A^. Instead of a 
mere radial line, may be substituted a point which at the same 
time traces the outline of a tooth space, so that a spherical screw 
thread is generated with which a spur gear will engage at any 
point. Fig. 632. If the axes A and B are maintained in their 
proper positions, the spiral when driven, will operate the gear 
in the same manner as a worm and worm wheel, | 221. 

The practical value of this especial form is extended by the 
fact that the axis of rotation need not coincide with a diameter 
of the circle. Under these conditions there may occur a num- 
ber of forms of bodies of revolution bearing an affinity to the 
sphere, and to which the writer has given the general name of 
globoids. The corresponding spirals may be called globoid 
spirals and the resulting gears, globoid spiral gear wheels. Many 
of these may be made of practical use. (See Fig. 633.) 

There are numerous forms of globoids according to the posi- 
tion which the describing circle holds to the principal axis. The 
axis about which the radius C S turns is called the counter-axis. 
It stands at right angles to the starting position of the describ- 
ing circle, and either intersects the principal .axis, or is inclined 
to it without cutting it. We have then r, for the radius of the 
describing circle ; a the shortest distance between the axes A 
and C, c the distance of the centre of the describing circle from 
the plane of the principal axis, (5 the angle which the principal 
axis makes with the plane of the describing circle, extending 
from 0° to 90°. This gives four classes of globoids, as follows : 

I. a^o, c = o. 
II. a^ 0, c chosen at will. 

III. a chosen at will, c = o. 

IV. a and c chosen at will. 

A right globoid is one in which 6 = 0, and when (5 is an acute 
angle we get an inclined globoid. 

The first class is represented by the globoid Fig. 634, giving a 
symmetrical conical section ; if J = o we obtain the previously 
described sphere. 



The second class gives the inclined globoid. Fig. 635, with 
unsymmetrical conical sections, with regard to the equator, the 
spiral being on the zone mantles. If c! = c we obtain a sym- 
metrical, cylindrical hollow section of a sphere, Fig. 636. The 
spiral, when a^o, becomes a spherical cycloid. If (5 = 90° the 
figure becomes a plane cone, or plane ring, and the curve be- 
comes a plane cycloid. 




Fig. 633. 

We have the third class when (5 ^ 0, and a > r, giving a so- 
called cylindrical ring, or right globoid ring. Fig. 637 a, and 
when a < r, the apple shaped globoid, Fig. 6373. 

If 6 is an acute angle, the globoid is flattened. Fig. 638 ; the 
globoid of Class I is the limiting case. The spiral curves are 
globoidal cycloids, which become plane figures when ^ -= 90°, 
and the globoid becomes a plane ring or plane cone. 




Fig. 635. 



Fig. 636. 



The fourth class gives the highest forms. Fig. 639, in which 
S ^ 0, and we may have a '^ r, a ^ r, or a <^ r. 

The inclined globoids of this class have forms, the limits of 
which are found in those of the secoud class. Fig. 635. If (5 = 
90° we have again the plane cone or plane ring.'- 

The practical applications of the globoid spiral gears are va- 
ried, and are found mainly in right globoids of classes III and 



* Two right globoid rings may unite to form a pair of machine elements 
when the thickness of one is made equal to the hole in the other, as in Fig. 
a. The two parts then bear the relation to each other of journal and bear- 
ing, and are similar to a balljoiut. Each of the two elements describes by 
the relative motion of any point a corresponding path en the other member. 




* More properly a spherical cycloid, see §216; its kinematic axoids are 
normal cones. 



These conditions are approximately found in a pair of chain links. Such a 
pair may also be considered as a contracted form of universal joint, A B C, 
Fig. b, the same relative motion existing between A and C. The same thing 
is shown in a fractional form in Fig. c, when some method of holding the 
parts together, such as bands, etc., must be used. This latter resembles 
closely the ball and socket joints of the human skeleton. ^ 



THE CONSTRUCTOR. 



143 



IV. In the valve geai" of Stephenson's locomotives, Fig. 640, is 
found a globoid worm of class III, using the middle part of the 
globoid apple, Fig. 637 b, (a <^ r). In this case the reversing 
lever B is really a part of an internal gear with a radius A\ = 
the radius r of the describing circle.* In this case the internal 
gear has but a single tooth, although more might be used. 




Fig. 637. 



It will be seen that the globoid forms can be used as internal 
gears. This is shown in Fig. 641, which represents a worm 
formed as a globoid screw. Its form is practically the same as 
that of the hole in the right globoid ring, Fig. 637 a. The sec- 
tion shown in the figure is of such length that it includes cue- 
fourth of the entire circumference of the worm wheel B, al- 
though it could be extended so as to include almost one-half. 




The most important point to Tae considered is the formation of 
the teeth. i?j is again made equal to r. Since the globoid is 
used in the internal form, the two tooth profiles, qn r and 7?, , 
fall together. The sliding is in the plane of a normal section 
through B and A A^ and not endlong, and hence the shape of 
the teeth is absolute. 




Fig. 640. 

(Internal gear tooth, with i? = 7?, ). The teeth can be made 
of straight profile in the worm wheel as well as in the worm.f 

The production of the globoid worm in the lathe is not diffi- 
cult. This form has been frequently used in recent work. The 
advantages appear to be in the simple form of tooth and in the 
completeness of the engagement. 



An interesting modification is that of Hawkins, Fig. 642*. In 
this case the wheel B is composed of friction rollers of quite 
large size and the friction is thereby greatly reduced. Instead 
of there being only four teeth, as would at first appear, there 
is in reality an ideal immber of teeth, a condition refeired to in 




Fig. 641. 

the fundamental discussion in ? 200. If for every revolution of 
the globoid screw, one tooth of the wheel engages, there must 
for each space formed between the rollers be 10 teeth to a quarter 
revolution, so that instead of 4 teeth in B, there are 4 (i -|- 10) 
= 44 teeth. 




Fig. 642. 

The gearing used in Jensen's Winch, Fig. 643, belongs to the 
globoid class IV, of the form shown in Fig. 639. Usually in 
this form a = r, although sometimes a < r, as in Fig. 639 c. 
7?j is again made = ;', and the internal globoid form used. The 
ratio is so chosen that a slow motion can be converted into a 

I 




* The worm and internal worm-wheel, Fig. 621, is another example of the 
preceding case. 

■j- This form is described by Snieaton as used in a dividing engine by Hind- 
ley, see also Willis. Principles of Mechanism, ist edition, 1S51J p. 163. 



fast one, as may also be done ^-ith fhe form shown in Fig. 641 
if the pitch of the worm is made sufficiently great The use of 
rollers instead of teeth makes a very satisfactory construction.t 

* Hawkins' Worm Gearing. Sci. Am. Supplement, No. T04, p. 1643. 

f See Uhland's Prakt. Masch. IConstrukteur, also Engineer, Vol. 24, p. 493, 



144 



THE CONSTRUCTOR. 



If in the first two classes of globoids the supplementarj- axis 
is removed an indefinite distance, the globoids become plane 
snrfaces, and the globoid screws thereby reach the limit. The 
limiting case of Class III is the ordinary worm and worm wheel, 
and another form is Long's spiral gearing, which also belongs 
to Class III ; a is chosen at will, c-=- o, 6^0. The globoid be- 
comes a plane cone and the globoid screw becomes an Archi- 
median spiral. If J? becomes indefinitely great we obtain a 
disk with a spiral groove engaging with a rack, the middle sec- 
tion having full tooth contact from top to bottom.* When this 
is brought into Class IV, we obtain the Archimedian spiral in 
its most general form, i.e., the evolute of a circle. 



whence : 



£. C.iLCULATION OF PITCH AND FACE OF GEARING. 



I 225. 

Pitch os Ge.a.r Wheels. 



Tooth Section. 



bt 



The dimensions of gear wheels must, for the same pressure 
on the teeth, be increased to meet shock in proportion to the 
increase in initial velocitj'. For slow running gears this action 
can be neglected. We may in this respect, therefore, divide 
gears into two classes, viz. : 

Hoisting Gears and Transmission Gears ; and includes under 
the term hoisting gears all those having a linear velocity at the 
pitch circle of not more than 100 feet per minute, and under 
transmission gears all those running at a higher velocity. 

For a pitch /, face b, length of teeth /, and base thickness of 
tooth //, we have for a tooth pressure /'corresponding to a stress 
S, the general formula; 

-.'4(I)(4J (-) 

and for the proportions of length and thickness already adopted 
we have : 

b t = 16.8 :^ (213) 

This assumes that the resistance of the teeth is proportional 
to their cross section, which is also equally true for those which 
have the same ratio o£ b to / to each other, a condition which is 
often of much service in practice. 

? 226. 
Pitch and Face oe Hoisting Gears. 
For a hoisting gear of cast iron let : 

(PR) = the statical moment of the driving force, 
.^= the number of teeth, 

R = its previously determined pitch radius, in inches, 
t = the pitch, 
we have for the given dimensions : 

t=o.2^ >/J_-_J_, -- = 0.073 V— ^— • . . . (214) 

/ = o.045 \-!-— ^, -^ = 0.0145 V -!___!_. . .(215) 
the face b being made 



R 



b = 2t (216) 

These are intended to give a fibre stress .S" of about 4200 
pounds. The actual stress is properly somewhat less, because 
the thickness of the tooth at the base is usually more than ^ t, 
as assumed in (213). 

P R 

Since the value of — y;— is the same as the pressure P, we can 

use (215) in cases in which Ponly is given, as for rack teeth. 

In discussing the preceding formulEe, consideration must be 
given to the elements which are usually given or selected in 
practice. 

Let t' and t be the pitch for two cases respectively, and Zand 
Z' the number of teeth. Also let S and S' be the strefs at the 

base of the teeth, and let the constant, 6 f — J \ ~r\ , which 

in (213) is made equal to i5.8, be called Cor C ; we then have, 
according to (214) : 



i= \ 2 71 



C{PR) 



SZ 



t ^ C S'' Z' ^' 



'7) 



and for the radii R and R' 



R^ 
R 



~ Zt ~ ^ C S' \zl 



(218) 



The value of C depends upon the ratio of the teeth, and upon 
the value of S for the material used. If we assume the latter to 
be the same for both cases, the number of the teeth alone re- 
mains to be considered. A reduction in the number of teeth 
increases the pitch, according to (217) ; and according to (218) 
reduces the radius. 



. 1. 16; 



so that the 7 toothed gear will be about 5^ as large as the ii toothed gear, or 
a 42 toothed gear for the same case would be about J+ as large as a 66 toothed 
gear, and with 1.16 times greater width of face. 



^.xainple i. 


Z = II, Z' = 7, hence 




f=NfT=N' 




i' = 2i'=I.l6^. 




-§=Nf^=N/ 




• • Fig. 644. 

The constant C, for a given series of gears, should be invari- 
able, and for ordinary spur gears may be taken equal to 16.S, as 
in (213). For the so-called ' ' thumb teeth, " (2 2 1 2), the constant 
may be much smaller, and hence permit an important reduction 

in dimensions. The value of --- for wheels of more than ten 

teeth is not less than 0.7, and introducing this value we get C 
^ S.4, that is o.s C; hence "thumb shaped " profiles are capable 
of sustaining twice as great a load as the ordinary form. 

Example 2. If, for a given moment {^P R) the thumb profile is substituted 
for the ordinary form, without reducing the number of teeth, the pitch may 
be reduced in the proportion 

i' =t -$/~^i = 79 t, 

or about o S times, with a proportional reduction in diameter and face. 

If. liowever, the teeth are taken in the above ratio of it : 7, we would have 
for the pitch. 



and the radius R' 




0,202 = 0.5S R. 



* See Civil Engineer and Arch. Journal, July, 1852, also Dingler*s Journal, 
Vol. 125. Weisbach, III, ist Ed., p. 449, »d Ed , III, 2, p. 87. 



The influence of the stress Sis always important, and it should 
not be increased above the normal value for the given material, 
■which latter is usually cast iron. An increase of one-fourth in 
the permissible stress would reduce the pitch and diameter only 
7 per cent., but on the other hand it must be remembered that 
too low a value of 6' causes an unnecessary increase in the size 
and weight, not only of the gears butalso of the bearings, frame 
work and other parts of the machine. The value of .5' used 
above, viz. : 4200 pounds, has been show in practice to give sat- 
isfactory results, and there appears to be no good reason for any 
great variation from it. 

When the gears are made of wrought iron, as is sometimes 
the case, 5 may be made much higher, and may indeed be taken 
double, say 8400 pounds. This gives a reduction in t' in the 

proportion of /^^o.5 = 0.79 /. 



THE CONSTRUCTOR. 



145 



Example 3. For comparison between a wrought iron gear of 7 teeth of 
thumb shaped outline, with a cast iifcu gear of 11 teeth of ordinary shape, 
we have : 



R' = 



0.5 :■: 0.5 



f:.)=-^ 



\^ o. loi = 0.47 R^ 



■ > 4 7 



' t \K 0.393 = 0-7 t. 



amdy = 0.7 b. 

In Fig. 644 the five cases given in the last three examples are 
shown on the same scale, side by side. In order to indicate the 
fact that the moment {P R) is the same in all cases, the shaft 
diameter has been shown. It will be apparent that there is no 
definite relation between the diameter of the shaft and the ra- 
dius of a gear. 

The invariability of the moment, which has been maintained 
in the preceding examples, does not exist of the tooth pressure 
Pupon the driven gear is again transmitted through a second 
so-called compound gear. If the pinion of a radius R, driving 
a gear R', compounds by a pinion R.^ on the same shaft into a 
rack -R'/i f°^ example, with a given pressure P, we have from 
(2«4) 

^ = Const. V-^, '-^f 



whence 



This gives 



R'^R 



4J£ ?Jl 
^ S' Z 

' ^ ^ R, C S' Z' 

4 



(219) 



R., 



a 
■ c 

Z,' t,' 



s_ 



zi 



But R.j. = Z-^i and R^' = Z./ //, and from formula (215) ; 
t'-t J^~I- 



C V 



Hence we get : 
t 



l-slf 



S_ Z/ ^ 
S'' Z' Z'' 
By selecting the number of teeth we may make 
Z,' Z' 



z.,. 



and then obtain ; 



^ = v/^' 



c 



s_ 

S' 



and for the radii : 



R^ 
R 



Z 



^ c 



S' 



(220) 



(221) 



Example^. A rack with a tooth pressure /", gearing with an 11 toothed 
pinion, is driven by a larger gear which again engages with an ir toothed 
pinion, Fig. 645, the teeth being of the usual sliape, and the material cast 
iron. 

This is to be replaced by making all parts of wrought iron, and reducing 
the number of teeth in the rack pinion to 4, as shown in g 212, all teeth being 
also altered to the thumb-shaped form. We then ha ve C'= 0.5 C, y = 2 5", 

and hence : t' = x/lT = J^ '. and R' = R -t^ \/ y^ = -^i R. 

It will be noticed that in this case the ratio between the larger 
gear and the pinion on the same shaft is such that in (217) and 
(218) both are determined for the same moment {PR.) 

Example 5. If, in order still further to reduce the dimensions, steel is used 
instead of wrought iron, thits parinitting a stress of 14,000 pounds, we have 
i' = jl v/ 0.5 X 0-3 = °-387 t, and R' = 0.4. jij R = 0.145 R, or about J R. 

The proportion of the results of the last two examples is 
shown in Fig. 645. The force P on the teeth of the rack is the 
same in all three cases. 

The statical moment on the main shaft is, however, reduced 
with the reduction in R', as is comsequently that of the inter- 
mediate shaft. 

The advantages of steel as a material for gear wheels have al- 
ready been referred to in ? 222. Its greater strength enables 
much lighter wheels to be used for the same service, than with 
cast iron. For a gear of cast iron and of steel, to act against 
the same moment, all other things being equal, we have, taking 

(' j^' ,. — 2 

5'== 14,000 and S:=42oo — , and also — '^0.3 = about — in 

t R 3 

favor of the steel. 

This gives for the ratio of weight [fiY, that is 0.3, the same as 
the ratio of S to S' , or say three to one. This advantage also 
exists for transmission gearing, although not to the same extent. 

If the velocity ratio in a compound train is comparatively 
great, it is interesting to note that the most advantageous ratio 



between gears lies between i : 9 and 1 : 10, this giving a miav- 
mum of shafts and of teeth.* 



k ^^K. 



SiV?i 



i^K 



-m-i 



Fig. 645. 

'i 227. 
Table of Cast Iron Hoisting Ge.a.rs. 



t 


p (^^) 

R 


PR 
Z 


t 

IT 


R 


(PR) 
Z 


% 


127 


10 


o.is 


107 


8.67 


H 


200 


20 


0.20 


190 


20.56 


¥r 


2S7 


35 


0.25 


297 


40.16 


n 


391 


55 


0.30 


428 


69.40 


I 


511 


82 


0.35 


583 


110.20 


iX 


798 


160 


0.40 


761 


164.50 


^'A 


1 150 


277 


0.45 


963 


186.00 


^H 


1564 


440 


0.50 


1020 


320.50 


2 


2044 


65 8 


0.60 


1712 


555-20 


2A 


3200 


1284 


0.70 


2330 


S81.70 



Example i. A force of loo pounds is exerted on a hand crank of 15 inches- 
radius ; what should be the pitch and face of a 10 toothed pinion for the fur- 
ther transmission? 

Here we have -^r- = ~ = 150, and the nearest value in the table 

Z 10 

in the third coUimn, is 160, which corresponds to a pitch of ij^ inches. The 

face is = 2 rf = 2j^ inches. 

Example 2. A rack is to work with a pressure of 2000 pounds on the teeth. 

This would give a pitch of about 2 inches, or as given in the 4th and 5th. 

columns, a pitch t : 0.65 tt, which is practically the same, and the width of 

face = 2 I. If the rack is made of wrought iron, we have t = 0.707 X 2 = 

1,414", and the face = 2.S''. 

^6 22S. 

Pitch and Face of Gearing for Transmission. 

The fibre stress S, which is exerted upon the teeth by the ac- 
tion of a given force P, should be taken smaller for transmis- 
sion gears as the circumferential velocity Z' increases, since the 



*If (i> be the total ratio, and /c the number of pairs of gears, and the ratio 
Z k 

between each pair be .ar = — we have qb = jr . The total number of teeth 

:u the train, _j. = ^ (Z + Z') = A Z' (i + x). Now k = -. , and the product of 



the number of teeth and the number of pairs gives 

[Inx]-^ 



I w . 



yk = - 



Diffenentiating and making the differential coefficient equal to zero we- 

get I n X = • — ^ which equation is satisfied by x = 9. 19- For example 

fj) = 600, and the number of teeth in smallest pinion = 7. We have the fol- 
lowing combinations: 

(a) ^ = 20, 30, gives jj/ = 7 (2 + 20 + 30) = 364, jy k = 728. 

(d) ^ = 4.5.5.6, gives jf =7 (4 +4+5+5+ 6) = 168, j'k = 672. 

{c) (/) = 6.10. 10, gives^ = 7 (3 + 6 + 10 + ic?) = 2Qg,_y k = 609. 

The last solution is the best, for although it requires m-ore teeth than (5>- 
it has one less pair of gears, and for solution {a) the number of teeth, viz.s 
210 is inconveniently great. 



146 



THE CONSTRUCTOR. 



dynamic action of shock and vibration also increases. For cast 
iron we may take : 

9,600,000 

V + 2164 

in which v is the lineal velocity in feet per minute. For steel 
^may be taken 3'/^ times, and for wood j'^j times the value thus 
obtained. For cast iron we obtain, for : 



S = 



(222) 



v^ 100 I 200 I 400 I 600 I Soo I 1000 I 1500 I 2000 I 2500 
5=4240 I 4060 I 3744 I 3473 I 3238 I 3034 I 2620 I 2302 I 206S 

For Steel : 



The velocity v may be obtained when // and R (the latter in 
inches) are given, by the following formula : 

v = = 0.5236 jV « {223) 

The selection of a proper value for v will be discussed below. 

It is also found that the breadth of face (5 should increase with 

the increase of P. Tredgold states that the pressure per inch 

P 

of face, that is — should not exceed 400 pounds. This, how- 
ever, is not to be followed implicitly, since pressures as high as 
1400 pounds have been successfully used in practice. It is bet- 
ter, however, to consider the question of wear from the product 

P . 
of -^ into «, which should not exceed a predetermined maxi- 

. P 
inuni. It is found that if-;- X « exceeds 67,000 the wear be- 
b 

comes excessive. In a pair of wheels where the teeth of both 
are made of iron, the greatest wear comes upon the teeth of the 
smaller wheel. In this case we may make 



— , — = not more than 28,000 
b 



(224) 



and if possible it should be taken at less than this value. For 
smaller forces this constant, which we may call the co-efEcient 
of wear aud designate as A, may readil}' be made as low as 
12,000, and even 6,000, without obtaining inconvenient dimen- 
sions. When the teeth are of wood and iron the wear upon the 
iron may be neglected, as the wear comes almost entirely upon 
the wooden teeth For wooden teeth the value of ^ should not 
exceed 28,000, and is better made about 15,000 to 20,000.* It is 
impossible to give exact values in such constructions, and it 
must be left to the judgment of the designer as to how far it 
may be advisable to depart from the values obtained from exist- 
ing examples. 

It must be remembered that the different values of A do not 
appreciably affect the strength, but rather control the rapidity 
of wear. When sufficient space is available and a low value can 
be given to the co-efficient of wear, it is advisable to do so ; if 
this cannot be done, the co efficient which is selected will give 
an indication of the proportional amount of wear which may be 
expected. 

In cases where a number of wheels gear into one other wheel, 
it is better to take, instead of the number of revolutions of the 
common wheel, the number of tooth contacts, that is the pro- 
duct of the revolutions and number of wheels in the group. 

If 7? is given, as is often the case with water-wheels, fly-wheels, 
&c.. Pis also known, and since A can be chosen we have, tak- 
ing jVto be the horse power transmitted : 



P^n_ 
A 



63,000 



N^ 
^ 



16.8 /> 
*^ S'b~ 



T 



(225) 



hence from (213) for ordinary teeth, 

JA.Z_A_ 
S Ji 
and for thumb shaped teeth, 

_S.4_/^ ^ __8.4_^ 
S b S 11 

If, however, as occurs in many cases, R is not j^reviously de- 



(226) 



termined, the choice of the number of teeth Z is unrestricted. 
In such cases we have for the width of face b : 

396,000 N 

If we give to A the successive values 30,000, 25,000, 20,000, 
15,000, 10,000 and 5,000, we get the following numerical rela- 
tions : 

Commo7i and TJiitmb Teeth. 



Common Teeth. Thumb Teeth. 



Pii _^ iV__ 
30,000 "" R 



i = 



S^ 14, 1 1 2, 1 3, 520112,467 1 1 1,565] 10,782; 10, 103 18725 [7665! 6886 

And /or Wood: 
5 = 2544 I 2436 I 2246 I 20S3 I 1943 I 1820 I 1572 I I3S1 I 1240 b: 



b = 



_Pn^ 
25,000 

20,000 

_Pn^_ 
15,000 

Pn 



3-iS 



= 4.2 



1\_ 

r'' 

N_ 

~r'' 

N 



R 



N_ 

'"z'r 



...Sg;. 



--26.4 j-y, t. 



504,000 

~r5^ 



252,000 

= ~75~ 



420,000 210,000 

n S ' 1! S 



•J ■! 6,000 i5S,oooj 
^-' . f/ —- I ' 

n S ' n S 



252,000 126,000 , 

?/ 5 ' n S 



) (227) 



N 



A'' 

b = = 6-3 -pj = 39.6 1=-. ; / = 

10,000 R Zt 



1 68, 000 
11 S 



■\i'-- 



b-- 



_Pjt.^ 

5,000 



N N 

^13.6- = 79.2^^; 



84,00c 
;7T~ 



^' = 



84,000 
42,000 



For transmission gears the minimum number of teeth should 
not be fewer than 20, in order that the unavoidable errors of 
construction shall not cause excessive wear ; for quick-running 
gears it is desirable to have still more teeth. The gear wheels 
on high speed turbines seldom have fewer than 40, aud often as 
many as So teeth. When wood and iron teeth are used, the 
least wear is produced when the wooden teeth are on the driver, 
because the action begins at the base of the tooth and passes 
toward the point, while on the driven gear the action is reversed. 

If desired a number of teeth Z can be calculated which will 
give a desired ratio b : L If we combiiie formuhe (225) and (226) 
we obtain the useful relation : 



396,000 
Y6.S^A~^ 






n'- S' N 



(0 



(22S) 



This shows the important influence of A upon Z, and the ef- 
fect of the number of teeth upon the wear ; also the important 
relation of the tooth profile, since the constant 16. 8 (or for 
thumb teeth S.4) appears in the second power. It is also seen 
that Z is dependent on the square of ii, and the square of S, 
other things being'constant. These points indicate the methods 
of obtaining the least stress. 

The value of — is sometimes made as great as 5. For wider 

faces and sometimes for narrower, the rim of the gear is made 
of two adjoining parts. 

Example i. — A water wheel of 60 horse power, 26 feet, 3 inches 111 diameter, 
moving with a velocity at the circnmference of 256 feet per minute, is to be 
provided with an intental gear wheel, the pitch circle bein^ 16 inches less 
radius than that of the water wheel, and gearing into a pinion which is to 
make 40 revolutions per minute. 

256 

AVe have : « = ^^-^ — , _^ __ = 3- 1 



and 



"1 



40 



also 



n/ 



3.14 + 26.25 

256(157-5— i6_ 



230 ft perininute. P- 



33000 X 60 



3-1 157-5 \ 230 

= 860S lbs. This gives a permissible stress S = ^loo lbs. nearly. We tfiII 

choose for the smaller wheel — ^ = 25,000, which gives —r- = — — = 
o n-y 

2SOOO , P 860S 

~ = 625, hence = r— = - — = 13^+ ■ "^^'e then have from (227) / = 

40 02.5 62.5 ^ ''' 

420,000 ,,, „, ., , „ ^ TT R 2JTI4I.5 

-i-^ = 2.56". We then have Z= —— = — -^ = 347. if we make 

40 X 4100 t 2 56 

34S teeth the wheel ma}' be divided into 12 segments of 29 teeth each. For 



the driven wheel we have Zi = 



"I 



^=— X348 = 
40 



27, whence R^ =. 



27 X 2.56 



2 IT 
= II". 

E-xaviph 2. — A turbine water wheel of 100 horse power has a vertical shaft 
making q6 revolutions per minute, and it is required to drive a horizontal 
shaft at 144 revolutions, hence a pair of bevel gears are required. We will 
select wooden and iron teeth, and let the wooden teeth be on the driver. We 
will assume v to be between 1200 and 1400 feet per minute, which gives J S 

= 1600, and make A = 25,000, also — p = 3- We then have from (22S) Z = 

■506,000 962 X i6oo2 X ICO 

^ J,\, TT ■ ■ = 70- we tnen have ifj = ^- .70 = 47 • 

16. S2 X 250003 3 ' 144 ' ^" 



* See case 10, in g 229 seg. 



also i 



420,000 
96 X 1600 



We then have Zi 
-- 2.73" sa3' '2}'^", 5 = 3/--= ^li", V = 1536 feet per minute. 



THE CONSTRUCTOR. 



H7 



F.xample 3. — In a giveu train of gearing, Fig'. 646, in which the correspond- 
ing wheels of both pairs are of the 
same size, the force transmitted in 
each case i? inversely as the number 
of revolutio'ns. In order to have the 

Pit 
co-efficient of wear x" alike in both 


cases it is only necessary to make all 
the gears of the same face. An ex- 
ample of this kind maj' be found iu 
the back gearing of many lathes. 

Example 4. -Let it be required to 
construct a pair of durable gears of 
wooden and iron teeth under the fol- 
lowing conditions: iV= 5, ?i = n-^ = 

60, and — = 2. We may make v = 

500, which gives, from {222), ^ = 2160, 
and as great durability is required 




Fig. 646. 
■we will take A as low as 10,000. These values iu (228) give 



396,000 6o2 X 2160- X 5 



16.8- X g,ooo'* 



which we may call So teeth. 
We have from (227) 



■■ 1. 167" 



and 

or 2 /, as intended. 

Example 5. — Let N = 40, n = 

of common form, and let -— = 

take A = 25,000. This gives for Ihe driver gear : 



— = 2.33 
gooo 80 X 1.167 

30, n-i = 50 for a pair of iron gears with teeth 



If we make v = 300, S = 3400 and we 

driver gear : 

396,000 5o2 X ^4oo2 X 40 

= 41.5 

70> 



i6.S2 X 25,000= 




2.5 




say 42 teeth, and Z-^ 




Z = 


we have t = 


420000 




50 X 


3400 





and ^ = 2.5 z! = 6.175". 

If we choose the thumb-shaped teeth, and make — - = 3.5 we get : 





120 


Z 

ai 




396,000 




55L 


X 3400= 
3-5 
210,000 


X 


40 


say 


S.4 
\&Zx 


- X 25,000= 

= 200. f 


1.2 









_._ .1 = 4.32. 
50 X 3400 

This gives smaller teeth, but larger radii than when the common form is 
used. 

When steel is used for gear wheels, special proportions are obtained. It is 
not too much to say that the value of the co-efftcient of wear A should be 
taken twice as great as for cast iron. The stress S, however, may be taken 
3^ times that permissible for cast iron. Taking these points into considera- 
tion in formula (228) we see that A would reduce the number of teeth by J-g, 

and S would increase it by ( 1 , that is, about n times, so that the net in- 
crease would be -Q— , if the above values are accepted. It may therefore be 

laid down as a rule that steel gears should have more teeth for the same ser- 
vice than cast iron gears. The ratio of face to pitch may be made quite 

large, and iu the case of double spiral gears (as Fig. 637) the ratio ■ — is some- 

T 

times made as great as 7 or 8. If the formula for thumb teeth be used, in- 
stead of the usual shape, the constant 16.S will give satisfactory results. The 
value obtained for the pitch is that for the normal pitch t = t sin -y, but the 
width of face is the actual width, as /', in Fig. 657, 2 t>' in Fig. 62S. 
Example 6. — Suppose the wheels given in Example 5 to be made with 

■double spiral teeth of steel. We take A = 56,000, and — =6, also 5= 12,800. 

7" 

We then get : 



396,c 



5o2 X 12,800- X 40 



3.42 X 56000- ' 6 

8.4 X 56,000 



= 87 



We have t = 



also b - 



i2,Soo X 50 



= 4.4" 



56,000 87 X 0-74 
If we take Z\ = 84, we get Z = 140 and d = 4%"- If ^ = ^o" we have 

Eia6o 0.866 ^^ 

We may take / = 0.375", which gives t = 0.866 X 0.S75 = 0.757" and 
^ _ 4.3 ^ 

T 0.757 

We have then finally J?i -= 11. 6", I? = ig.47" 



= 5.93, or nearly 6. 



i 229- 

Examples and Comments. 

The following examples taken from actual practice will be of 
interest : (see Table on following page). 

No. I. Prom tlie driving gear of the main steam engine of 
Fleming's Spinning and Weaving Mill in Bombay. The toothed 
fly-wheel is the driver, and the teeth are shrouded, as shown in 
Fig. 651. The coefScient of wear for the driven gear seems high, 
and does not indicate long endurance. 

No. 2. A toothed fly wheel engaging with a pair of equal spur 



gears ; 300 horse-power transmitted by each gear, making a 

P >i 
total of 600 horse-power. The value for — — must therefore 

be multiplied by 2 ; see last column of the table. 

No. 3. This is from the air compressor for the atmospheric 

P71 
railway of St. Germain (now abandoned.) — - — is evidently too 

high, as would probably have become apparent had the gears 
continued in operation. 

P . 
No. 4. — - IS very high, but the small number of revolutions 


P u 
keeps the value of — — within reasonable limits. 

N0S.5 and 6. These are from the great water wheel at Greenock. 

The pressure at the rim is great, but the teeth have worn well 

in practice, as might have been predicted from the moderate 

Pn 
values of — : — The value of the latter is almost the same for 
o 

No. 6 as for No. 5, hence the wear should be about the same for 
both gears. 

No. 7. The teeth iu the smaller gear are thinner than those of 
the large fly-wheel, hence the two values for 5'. Probably the 
larger wheel was originally made with wooden teeth. 

Pn 

No. 9. Notwithstanding the high pressure the value of 

;/ 

is reasonably small. The stress upon the teeth is quite high, as 
is also the case with No. 4, and lower stresses are to be recom- 
mended. 

No. 10. This is one of the most noteworthy examples of the 
whole collection, on account of the very slight wear exhibited. 
The wooden teeth on the large wheel, (the fly-wheel of the 
steam engiue of the Kelvindale Paper Mill at Glasgow) ran for 
26}i years, for 20 hours per daj", with a wear upon the teeth, 
measured at the pitch circle, of only about ji inch. For the 
first half of this time the engine indicated 84 horse-power, at 38 
revolutions. The teeth were lubricated twice a week with talc 
and graphite. The long endurance is doubtless partiallj' due to 
the great care which the teeth received , they having been cut upOn 
the wheel in place, but also to the moderate.co-efficientof wear. 

No. II. The teeth were found too small in practice, as is indi- 
cated by the stress of 3000 pounds ; from formula (222) we ob- 
tain S= 1734 pounds. 

No. 12. Two gears with wooden teeth engage with a single 
pinion on the screw propeller shaft. The teeth are in two sets 
of 4:14" width of face each. 

No. 13. Very high pressure, Which must appear in the wear 
upon the teeth ; apparently it should be difficult to keep them 

P 
in good condition, owing to the high value of ---. 

No. 15. These teeth appear weak, as has been shown by re- 
peated breakages. The wear must be rapid, as indicated by the 

Pn 
high value of — , — . 
° d 

No. 17. These gears, (designed by Fairbairn) w^ere intended 
ultimately to transmit double the power at first given, iu which 
case the stress would reach over 4000 pouuds, which is admissible 

P/i 
but the value of — : — would then become rather too high to in- 
dicate very great endurance. 

teeth ; it is almost too great also for the iron teeth, and it must 
be remembered that with wooden and iron teeth, the wear comes 
almost entirelj' upon the wooden teeth. 

No. 22. These gears are from an establishment which has used 
hyperboloidal gears with much success for power transmission. 
The angle of the axes is 90°. The use of wooden teeth upon the 
driver is to be criticised, as tending to increase the liability to 
wear. 



No. 20. The value of 



■ seems too high for the wooden 



r. THE DIMENSIONS OF GEAR WHEELS. 
I 230- 

The Rim. 

The ring of metal upon which the teeth of a gear wheel are 
placed is called the rim. For cast iron spur gears, the thickness 
of the rim is given by the formula 

(f = 0.4 / -f 0.125" (229) 



148 



777^ CONSTRUCTOR. 





EXAMPLES OF TRANSMISSION GEARING. 


No. 


N 


n 


R 


Z 


t 


6 


V 


P 


S 


P 

b 


Pn 

b 


REMARKS. 


1 


1000 


36.67 
1 14.8 


3S.25 


144 
46 


5-25 


24 


2300 


14,000 


1877 


5S3 


fi'39? 
66,970 


Iron and Iron. 
Steam Engine. 


3 


300 


100 


I46.S 

37 


230 

58 


4.00 


14 


1900 


5,100 


1614 


364 


2x9107 
36,400 


Iron and Iron. 


3 


270 


60 
12 


19.6 
~9S^ 


_i9_ 
95 


6.25 


20.6 


616 


14,300 


1S4S 


694 


41,650 
S,330 


Iron and Iron. 


4 


240 


44 


no 


208 
68 


3-125 


16 


766 


10,200 


3270 


639 


_8,498 
28, 1 10 


Iron and Iron. 
Transmission for No. 8. 


5 


192 


15-14 


400 
3S"-25 


704 
62 


3-6 


15 


280 


22,240 


7252 


1483 


1,972 
22,450 


Iron and Iron. 
Water Wheel. 


6 


192 


50 


106 


208 

~6r 


3-i8 


15 


S40 


7,425 


2275 


495 


7,494 
24,750 


Iron and Iron. 
Transmission for No. 5. 


7 


140 


30 

55 


58^ 
32 


_L32_ 

72 


2.8 


8.6 


900 


5,000 


4266 

48:^5 


581 


17,440 
31,970 


Iron and Iron. 
Steam Engine. 


8 


140 


54-5 


66.5 
35-75 


133 
76 


3 


13 


1045 


4,350 


3700 


335 


10,040 
18,230 


Iron and Iron. 
Steam Engine. 


9 


120 


1-51 

13-3 


_29I 

33 


560 
80 


3-125 


15 


240 


16,230 


5688 


1082 


1,634 
14,390 


Iron and Iron. 
Water Wheel. 


10 


100 


45 
158,8 


24 


J76_ 
50 


3 


10 

5-9 


2000 


1,635 


924 


163 


7,357 
8,175 


Wood and Iv-on. 
Steam Engine. 


11 


90 


26 

80 


S5-4 
27-75 


228 
74 


2-375 


1 163 


2,500 


3000 


424 


11,010 
33,900 


Wood and Iron. 
Steam Engine- 


12 


82.5 


54 
83 


55- 1 
35-S 


114 
74 


3.1 


2X4.7S 
11.75 


1558 


3,440 


1848 


362 


19,540 


Wood and Iron. 
Screw Steamship. 


2x30,040 


13 


50 


4.o_ 
7"32 


50-4 
27.5 


96 

52 


3-25 


10.6 


104 


15,500 


7536 


1463 


5,849 
10,700 


Iron and Iron. 
Water Wheel. 


14 


20 


_Z:Zi 
40 


85.4 
16.5 


24S 
48~ 


2.2 


6-3 


328 


1,980 


2420 


314 


_2,433 
12,570 


Iron and Iron. 
Water Wheel. 


BEVEL, GEARS. 


15 


300 


50 


24-37 
45-7 


50 
93 


3-1 


13 


11S7 


8200 


3270 
3697 


630 


58,660 
31,540 


Iron and Iron. 
Turbine. 


16 


300 


100 


29.7 
26.7 


_55_ 
49 


2.7 


10 


1576 


6170 


3840 


617 


61,700 
68,980 


Iron and Iron. 
Transmission for No. i. 


iii.S 


17 


240 


44 

44 


42 


75 


3-5 


iS 


96S 


S050 


2133 

2000 


447 


19,670 


Iron and Iron. 
Transmission for No. 3. 


18 


200 


41 
80 


30.1 


98 
50 


3-8 


II. 8 


1260 


5157 


437 


17,920 
34,960 


Wood and Iron. 
Turbine. 


19 


130 


124 


31-3 

24.S 


So 
60 


2.4 


8 


1523 


2772 


2276 
2417 


346 


32,220 
42,970 


Wood and Iron. 
Turbine. 


20 


100 


93 
144.7 


*3-4 

15 


_7^ 
45 


2.1 


6.3 


1 140 


2860 


2985 

3840 


454 


42,220 
65,690 


Wood and Iron. 
Turbine. 


21 


50 


93 
218 


25.6 
10.8 


_J5_ 
32 


2.1 


6-3 


1236 


1313 


1564 
1 848 


20S 


£9,380 
45,430 


Wood and Iron. 
Turbine. 


HJTPERBOLOIDAIi GEARS. 


22 


16 


72 
81.6 


21.6 
19 


68 
60 


1.996 
1-993 


5-9 


S12 


640 


2il loS 
1250 


7,Sio Iron and Wood. 
"8^851 Transmission. 





THE CONSTRUCTOR. 



149 



See Fig. 647. The rim is thickened in the middle or at one 

edge to — ^, and also stifiened by a rib, and for gears of fine 






l,5rfi 



Fig. 647. 

pitch the section of the rim is curved, which harmonizes well 
with arms of oval section. According to (229) a pitch of \" 
would give a rim thickness <! ^= 0.4" -\- 0.125" = 0.525" or a 
little over yi", and for a pitch of yi" , 6 = 0.325". 

For bevel gears of cast iron the rim is made — S thick at the 

outer edge, and of the various forms shown in Fig. 648. 








Fig. 648. 

For wooden teeth it is necessary to have a deeper and stronger 
rim, the dimensions being dependent somewhat upon the 
method of inserting the teeth. The proportions for spur gears 





Fig. 650. 



are shown in Fig. 649, and for bevel gears iu Fig. 650. For very 
wide faces the wooden teeth are made in two pieces and a stay 
bar cast in the mortise. 

Small pinions are often cast solid, and when subjected to 
heavy pressures are strengthened by shrouding, as shown in 
Fig. 651, and sometimes this shrouding is turned down to the 
pitch line. 






Fig. 651. 

For double spiral gears of steel (see \ 223) shrouding is to be 
recommended, and is verj' generally used. The use of shroud- 
ing especially assists in securing good steel castings, for the 
great shrinkage of the steel, nearly two per cent., tends to pro- 
duce warped and twisted castings. 

Small pinions are sometimes cut from solid wrought iron, iu 
which case the shrouding must be omitted. 



Thb Arms of Ge.^r Wheels. 

The arms of gear wheels are made according to the following 
forms, dependent upon the kind of rim used. 



/S;.::l 



1r 



o,sJ 



Pi 



-j:::^- 



JS3 




0,8 <J 
Fig. 652. 



Fig. 652. Ribbed sections, which are made sometimes as 
shown in the dotted lines as may be most convenient in mould- 
ing, and oval sections, in which the thickness 1} of the arm is 
generally made one-half the width h. A good proportion for 
the arms is obtained when their number A is made as follows : 



^ = 0.55^^^^ 
= 0.73/771 



A 



(^30) 



From these we obtain the following : 



A 
Z 



i 
f 



4 



5 



36 



6 
119 

52 



7 

162 

71 



475 



93 146 209 



Example. — For a gear wheel of 50 teeth and 2" pitch, we have Z \/ t =" 
50 \/ 2 = 50 X 1-414 = 70 and this lies between 53 and S3; being nearer the 
latter we give the wheel five arms. If the pitch had been %", and the same 
number of teeth Z \/ ^ = 50 v^o.75 = 50 X 0.S66 = 43.3 or between three 
and four arms, the latter number being used in practice. 

The width of arm /;, in the plane of the wheel is somewhat a 
matter of judgment, but may suitably be made according to the 
ratio A = 2 to 2.5 t, when the thickness ;3 may be obtained from 
the following formula : 



= 0.07 



Kl) 



(231) 



Should this formula give 
small for convenience in casting, another value for 



thickness either too great or too 
■— must be 



taken and the calculation repeated. The following table will 
assist in this operation. 

The taper of the arms may be made as follows : the ribs at 
the rim are made slightly narrower than the breadth of face b, 
and at the hub, equal to, or slightly greater than b. For arms 
of oval section /;, may be made equal 2 / at the centre of the 
wheel, tapering to 73 this width at the rim. 

§ 232. 
Table of Gear Wheel Arms. 



h 
1 


9 
Value of ^, when 
b 


Z 


9 


12 


16 


20 


25 


3° 


35 


40 


I-50 


0.20 


0.28 


0-37 


0.50 


0.62 


0.78 


0.93 


i.oS 


1.24 


■•75 


0.16 


0.21 


0.27 


0-37 


0.46 


0.57 


0.69 


o.So 


0.91 


2.00 


0.12 


0.16 


0.2I 


0.28 


0-35 


0.44 


0-53 


0.61 


0.70 


2.25 


o.ro 


0.12 


0.17 


0.22 


0.28 


0-35 


0.41 


0.4S 


0-55 


2.50 


o.oS 


o.io 


0.13 


0.18 


0.22 


0.28 


0-34 


0-39 


0-45 


2-75 


0.06 


ooS 


Oil 


0.15 


o.iS 


0.23 


0.2S 


0.32 


0-37 


3.00 


0.05 


0.07 


0.09 


0.12 


o.i5 


0.19 


0.23 


0.27 


0.31 



ISO 



THE CONSTRUCTOR. 



Example. — Let a wheel have 6 arms, and 120 teeth of 2 inch pitch, t}ie face 
being 4 inches. If we make /^ ^= 2/ at the centre of the wlieel, we have 

-— — = 2, and — ^ = 20, heuce we get from the table — y— = 0.35, and/3 = 4 X 

0.35 = 1.40". If this is considered too thick, we ma}' make h = 2.25/, which 
grives ^ = 4 X 0-2S = 1. 12". 

For gears with wooden teeth, and for the iron wheels gearing 
with them, the dimensions of the arms may be made o.S times 
that given by the preceding rules. If more accurate dimensions 
are required, the best plan is to determine the pitch of the 
equivalent iron teeth, and use this value in the calculations. 

\ 233- 
Gear Wheel Hubs. 
The hub for a gear wheel generally tapers slightly each way 

jfrom the arms to the end, the length Z = — b, or somewhat 

4 
more for wheels of very large diameter, and the thickness of 
metal about the bore is made 'w ^= 0.4/?. 4- o-4", in which h is 
the same as in the preceding section. In cases of much im- 
portance reference should be made to formula (66), \ 65. 

If the wheel is not to be secured by shrinkage the thickness of 
metal at the ends of the hub may be made = 3/ lu. The key way 
is ctit the entire length of the hub, and for wheels which are 
subjected to heavy service the metal should be reinforced over 
the key way. Instead of this, the hub may be strengthened by 
wrought iron rings, forced on one or both ends. Such rings are 
nsuallj' of rectangular cross section, the thickness being yi iv, 
aud add greatly to the strength of the hub. See Chapter III. 
g 161 to the end. 

? 234. 

Weight of Gear Wheels. 

The approximate weight G of gear wheels proportioned ac- 
cording to the preceding rules may be obtained from the fol- 
lowing : 

G = 0.0357 b I- (6.25 Z + 0.04 Z'-) . . . . (233) 

The following table will facilitate the application of the 

Q 

formttla as it gives the value of —. — — for the number of teeth 

which may be given, and the weight can at once be found by 
multiplying the value in the table by b P. 



z 





2 


4 


6 


8 


20 


5.04 


5.60 


6.1S 


677 


7.3s 


30 


7-99 


8.6i 


9.24 


9.S9 


10.52 


40 


11.09 


H.90 


12.59 


13-30 


14.02 


5° 


14.74 


I5-4S 


16.23 


17.00 


17.77 


60 


1S.55 


1935 


20.15 


20.97 


21.80 


70 


22.65 


23.50 


2436 


25-24 


26.12 


So 


27.02 


27.93 


2S.85 


29-79 


30.73 


90 


31-69 


32.66 


33-63 


34-62 


35-63 


100 


36.63 


37-67 


38.70 


39.75 


40.S1 


120 


47.40 


4S.54 


49-69 


50.S5 


52.03 


140 


59-3° 


60.56 


61.82 


65.10 


64-27 


160 


I?-^5 


73-73 


75.10 


76.39 


77-90 


180 


^Ht 


8S.01 


89.5;. 


91.02 


92-54 


200 


101.88 


103.48 


104.98 


106.70 


108.34 


320 


11S.36 


120.08 


122.15 


123-52 


125.27 



ExampU.—Voz a cast iron gearwheel, proportioned according to the fore- 
going rnles, with 50 teeth, 2'' pitch and 4" face, we have b C- = 16, and by the 
table the multiplier for 50 teeth is 14.74, and the weight = 16 X 14.74 = 235-S4 
lbs., say =36 pounds. For a gear of 50 teeth, i>.;" pitch and 2)4" face, we have 
bi- = 3.90625, which multiplied by 14.74 gives 57.62 pounds. 

For bevel gears or for gears with wooden teeth and lighter 
arms (as given at the end of l 232) the weights will run slightly 
less than given by the table. 



CHAPTEK. .XVIII. 

RATCHET GEARING. 

i 235. 

Classification of Ratchet Gearing. 

Ratchet gearing may be considered as a modification or ex- 
tension of wheel gearing. The object of ratchets is to check the 
action of certain portions of a machine or train of mechanism 
and so modify au otherwise continuous motion into some inter- 
mittent form. 



Ratchet gearing may be divided into two main divisions 
according to the nature of the checking action. When the 
movement of the checked member is impeded in only one 
direction we have what may be called a Running Ratchet ; and 
when the movement is checked in both directions, a Stationary 
Ratchet. 

The distinction -will be understood b}' reference to the accom- 
panying illustrations, in which Fig. 653 shows a ratchet wheel 
aud pawl a b c, the shape of teeth and pawl permitting motion 
of the wheel in one direction, and hence forming a Running 





Fig. 653. 



Fig. 654. 



Ratchet Gearing, while in Fig. 654 the rectangular notches and 
pawl for a Stationary Ratchet Gearing. The lifting of the pawl 
is called the release, and the falling into gear is called the en- 
gagement of the ratchet gearing. 

If the two members b and c are held, a becomes the intermit- 
tent mover, while if a be held, the parts b and c possess the 
intermittent action ; as for example, the sustaining pawl and 
ratchet wheel of a common hoisting winch in the first case, and 
the reverse lever aud quadrant of a locomotive in the second 
case. 

Ratchet gearing is a portion of constructive mechanism which 
will repay close investigation. For this purpose the following 
six groups may be considered : 

1. Ratchets pure and simple, such as a ratchet wheel and 
pawl for the mere prevention of rotation. Examples: the 
ratchets of a windlass, or of the beam of a loom. 

2. Releasing Ratchets ; those which act to release members 
which are under stress, and which by such release are permitted 
to perform and deternd late work. Examples: the pawls which 
release the drop of a pile driver, the trigger of a gun, or the trip 
valve gear of some steam engines. 

3. Checking Ratchets ; those which arrest parts which are 
already in continuous motion. Example; the safety check 
.ratchets upon elevators, and upon mine hoists. 

4. Continuous Ratchets ; those in which a combination of 
pawls acts to drive a member in a given direction with practi- 
cally a continuous motion. Examples: a ratchet-driven wind- 
lass ; sotne forms of counters. 

5. Locking Ratchets ; those which act to detain certain mem- 
bers y- a fixed relation against the action of external forces 
until released. Examples: some forms of car couplings aud of 
releasing shaft couplings, also the mechanism of locks. 

6. Escapements ; those forms which permit a member under 
the action of an impelling force to make a regularly intermit- 
tent motion in one direction. Example: the various forms of 
clock and watch escapements. 

By following this classification, the various principal funda- 
mental forms may ^e briefly examined. 

I 236. 
Toothed Running Ratchet Gears. 

In running ratchets, the direction of motion which is not 
checked by the pawd is called the forward motion, and the re- 
verse, the backward motion. The teeth on the ratchet wheel 
must 'therefore be so shaped that when the pawd is in engage- 
ment the backward motion ouly must be impeded. It is also 
important that the form should be so chosen that the first ten- 
dency toward a backward movement should act to produce an 
engagement of the pawl wilh the teeth. 

In determining the fornj of teeth. Fig. 655, we observe that 
the most effective point upon the circumference of the wheel for 
the action of the pawl is that at which the joining line 1.2 of 
the centre of the wheel i, with the point of the pawl 2, is at 
right angles with the pawl radius 3.2. If we describe a circle 
upon the diameter 1.3, or the distance between centres of wheel 
and pawl, the intersections 2 and 2' with the pitch circle of the 
ratchet wheel will give the tv>-o most advantageous points of 
application. If the point 2 be selected, the attempted reverse 
movement of the wheel will subject the pawl to compression, 
while if 2' be chosen the pawl must be of the hook shape shown, 
and will be subject to tension. If the teeth of the wheel are to 
be of straight outline, the flanks should be radial. If a point of 



THE CONSTRUCTOR. 



151 



action 2, or 2,, in front or behind 2 or 



be chosen, the 



mechanism will be operative, but less advautageousl)' than when 
constructed as above, for the lever arm of the force-couple act- 




FlG. 55s. 

ing upon the wheel will be less, and hence the pressure greater. 
The angle of the flank, which will cause the direction of the 
force upon the pawl to pass through the axis 3, is found by 
erecting a perpendicular from 2^ or 2^ upon 2^ . 3 or 2^ . 3. 






Fig. 656. 

It is not necessary to bevel the end of the pawl so that it shall 
bear in but one point of the tooth, as it is not ditScult to shape 
the tooth profile so that the force /-"shall jjass through the axis 
3, when the pawl engages with the tooth. This is accomplished 
by making the profile of the flank of the tooth a circular arc 
struck from 3 as a ceutre, as in Fig. 656 a. 





Fig. 657. 



Fig. 658. 



The same result will be attained by giving this curve to the 
end of the pawl, and making the point of the tooth the bearing, 
as at b, or both pawl and tooth may be formed to the curve, as 






Fig. 659. 

at c. Since the force which acts upon the pawl has no tendency 
to cause it to lift out of gear, when constructed as thus described 



we may call this form of tooth the "dead" ratchet tooth. 
Other forms of teeth will be considered hereafter. 

Internally-toothed ratchet wheels may also be made with the 
pawls adapted to act either in tension or compression, as at 2 
and 2', Fig. 657. The axis 3 may be within the wheel, Fig. 65S, 
in which case the above given conditions for the best position 
of the point of action cannot be fulfilled. 

If the radius of the ratchet wheel be made infinitely great we 
have a ratchet rack, Fig. 659, in which a is a pawl acting in 
compression, and b a form acting in tension. 

An important application of the ratchet rack is shown in Fig. 
660, which is the upper portion of the lifting frame for a screw- 
propeller.* 




Fig. 660. 

The two ratchet racks a, which support the frame as it is grad- 
ually lifted, are in the middle plane of the ship, being fast to the 
walls of the propeller well. In order to insure the engagement 
of the pawls b b, they are held in geai by the loop springs of 
rubber. The frame is raised and lowered by a rope tackle, the 
sheaves of which are shown, the so-called " c/icvse-conpling''' 
(see I 156), permitting the propeller to be lifted, when its tongue 
and groove are in the proper vertical position. The pawls are 
held out of gear by means of lines, during the operation of 
lowering. The frame and ratchet racks are both made of bronze. 
The bent lever is another pawl which engages in a notch m a 
blade of the propeller, and prevents it from revolving during 
the operation of raising or lowering. There are two wooden 
struts, the bronze shod ends of which can be seen on each side 
just above the pawls b, their function being to hold the frame 
firmly in its lowest position, when the propeller is revolving. 





Fig. 661. 



Fig. 662. 



Ratchet racks are also used extensively in connection with the 
hoisting machinery in shafts of mines, etc. 



* See Fig, 323, § 117, -where one of the bearings for the same propeller is 
shown. 



S52 



THE CONSTRUCTOR. 



Instead of giving the ratchet wheel an infiuitel}- great radius, 
the arm 2.3 of the pawl may be made iufinitely long. This 
simply means that the motion of the pawl is guided in a straight 
line, in some form of slide. In Fig. 661 such an arrangement is 
shown for a ratchet wheel, and in Fig. 662 for a ratchet rack, 
such forms being not uncommon. 

I 237. 

The Thrust upon the Pawi<. 

The condition that the thrust upon the pawl, in a ratchet gear- 
ing, shall pass through the axis of the pawl, is not always ful- 
filled, and in some cases it is impracticable to attain such a 
relation of the parts. The mutual action of the pawl and ratchet 
-wheel upon each other must therefore alwa3'S be considered. 
If the flank of the tooth of a spur ratchet wheel (or a tangent 
"to the flank of the outline is curved) does not form a right angle 
■with the plane 2.3 of the pawl, there may exist, under some cir- 
cumstances, a tendency to force the pawl into the tooth, or in 
other cases to throw it out of gear. 



T,/ / 



.•■•■ .Ti Tj, 



••• Ni 



B-; 




Fig. 663. 

In Fig. 663 the various cases are examined. If at the point of 
contact 2 a normal iViVjto the plane of the tooth flank be 
drawn, this normal may bear one of three relations to the tri- 
angle 1.2.3. The "thrust-normal" N jVj may fall without 
the triangle, or within the triangle, or it may fall upon one of 
the sides of the triangle. 

If it falls upon 2 . 3, the thrust is neutral ; if it falls upon 2 . i, 
the thrust is zero ; that is, there will be no action of the pawl 
■upon the wheel, or vice versa,, barring the action of friction. 

The angle S between the line 2. 3 of the pawl and the tangent 
at 2, which is equal to the angle between the normal to 2 . 3, 
and the "thrust-normal," is called the angle of thrust. By 
considering this in connection with the angle of friction (p vari- 
ous relations are obtained. 

On the one part, the force applied will act to alter the posi- 
tion of the pawl, either to or from the centre of the ratchet 
•wheel : on the other part, it will also act to move the ratchet 
•wheel forward and backward. 

These relations are classified for various conditions in the fol- 
lowing table, in which a ^orce which acts to force the point 2 
from I is called an " outward " action, and the reverse, an " in- 
ward" action. 



For the so-called " dead " ratchet tooth a = 90°, case i, hence 
there is tendency neither to inward or outward movement. 
The variations above given are, however, more or less used in 
practice, and the table will be of service in considering the 
action in such cases. Some examples will be given here, and 
numerous others may be found in subsequent illustrations. 

In many cases it is desirable that the pawl should be held in 
engagement with the tooth by the action of the impelling force, 
as in Fig. 664, this falling under the fourth or sixth case. This 




Fig. 664. 

form of tooth insures the retention of the pawl in place after it 
has once entered the tooth, and is sometimes used in hoisting 
machinery when heavy loads are to be sustained ;_ au applica- 
tion is also found in Pouyer's Coupling, Fig. 453, in which the 
secure engagement of the pawls is an important point. 
Another secure form of pawl is shown in Fig. 655. 




Fig. 665. 

In this case the wheel is made with pin-teeth. The pawl has 
a forked end, the inner flank tending to produce an inward 
movement, the outer flank, outward movement. 

In this case, as in the preceding, the wheel must be turned 
through a small angle before the pawl can be released. 





ANGLE OF THRUST c = 90°. 


The Thrust 
Action is : 


The In^pelling Force : 


Outward Movement: 


Inward Movement: 


i) neutral. 


is without effect. 


is without effect. 


is without effect. 


ANGLE OF THRUST c<C90° and > 90° ^<p. 


2) inward. 

3) outward. 


is without effect, 
is without effect. 


produces reverse motion, 
produces forward motion. 


produces forward motion, 
produces reverse motion. 


' AIVGLE OF THRUST a<:90° — :p and > <i. 


4) inward. 

5) outward. 


produces inward mo\'ement. 
produces outward movement. 


produces reverse motion, 
produced by impelling force. 


produced by impelling force, 
produces reverse motion. 


ANGLE OF THRUST c<i<f,. 


6) inward 

7) outward. 


produces inward movement, 
produces outward movement. 


is without effect. 

produced by impelling force. 


produced by impelling force, 
is without effect. 


ANGLE OF THRUST a = o. 


S) null. 


produces inward movement. 


produces friction only. 


produces friction only. 


1 



THE CONSTRUCTOR. 



153 



I23S. 

The Si,iding Flanks. 

We have discussed the action of the flanks of tooth and pawl 
■which work together during the thrust. It is obvious that 
greater libert}' is permitted in the form of the sliding flanks. It 
is only necessary that the form shall be such that the forward 
movement of the ratchet wheel shall lift the pawls properly out 
of gear. The forms fall under cases 4 to 7. The usual form is 
the common zig-zag ratchet, but others are also used, as in 
Figs. 666 and 667, in both of which the teeth are symmetrical. 








-r-j: 



Fig. 666. 



Fig. 667. 



If it is desired to have the end of the pawl symmetrical, as in 
Fig. 667, this may be done, and the pawl may be reversed for a 
reverse movement as shown in the dotted lines. This form is 
used on the feed motion of some machine tools. 

For some purposes it is desirable to form the thrust flank 
upon which the impelling force acts, in the same manner as the 
sliding flank, in which case the pawl must be held in gear by 
some extraneous force capable of resisting the maximum im- 
pelling force which it is desired shall act. 




Fig. 668. 

Such a form is shown in Fig. 668, which is similar to the nut- 
locking device shown previously in Fig. 241. 



Spring Ratchets. Quadrants. 

The form of ratchet last described possesses an especial prop- 
erty, that is, the action of the spring tends to force the pawl 
into the space as soon as the point is over the middle of the 
tooth. This causes the pawl to spring into engagement, hence 
the name spring ratchet, and this action causes an acceleration 
of the motion either forwards or backwards as the pawl is forced 
into the space. Applications of this form are found in repeating 
watches, in which the wheel is star-shaped, and hence called 
the star, while the pawl is called the star pin or springer.* 




A modified form. Fig. 669, is used in Thomas' Calculating 
Machine. In this case the spring itself acts as the pawl, being 
attached directly to the arm without joint, forming a plate link. 
(See I 180.) 



Instead of using an entire ratchet wheel, a portion only need 
be made, if the required movement is but small, and in some 
cases reduced only to a single tooth, as in Fig. 670. 




Fig. 670. 



Fig. 671. 



Sometimes the two members may oe made of similar form, 
each working alternately upon the other. Fig, 671. Examples 
of this are found in the valve gear of some Cornish engines. 
These belong to the so-called "dead" ratchets, and are called, 
more or less appropriately, quadrauts, or sextants. 

I 240. 

METHODS OF Securing Pawls. Silent Ratchets. 

The engagement of the pawl with the ratchet wheel is usually 
secured by the weight of the pawl, sometimes assisted b}' addi- 
tional weights, as in Fig. 659. This may also be accomplished 
by means of a spring. It is desirable to give such springs but 
little movement, and small frictional resistance. It should 
therefore be placed near the axis 3, and is best placed in the 
line I . 3, so that 3 . 4,5, shall line in the same straight line, 
Fig. 672 a. If this cannot be conveniently done, it may at least 




Fig. 672. 

be made nearly so, as at b. A weak spring with much move- 
ment may be seen below in Fig. 6S0, yet at the same time the 
line 3 . 4 . 5, is only slightlj' varied from a straight line. In 
spinning machinery spiral springs of steel are used, and rubber 
springs have been used in propeller hoisting frames, Fig. 660. 

In devices in which the pawls are sometimes above and some- 
times below the wheel the springs are sometimes replaced by 
using several pawls. This is shown in Fig. 673, which is Wil- 
ber's ratchet for use in lawn mowers. 





Fig. 673. 



Fig. 674. 



♦This is shown later among the releasinjr ratchets. 



Three pawls, with half journal, are here used, and as the axis 
I, lies in a horizontal position some one of the pawls is always 
in engagement by its weight. The movement of the teeth 
under the pawl, and the dropping of the latter into the spaces 
produces wear upon the parts, and to avoid this action various 
devices have been made ; these being known as silent ratchets. 

A very useful form of silent ratchet is shown in Fig. 674. 
The pawl is made with a projection 5, which is connected to a 
friction band d, which is carried upon a hub 4 on the ratchet 
wheel. When the wheel begins to move forwards, the arm 4 . 5 
lifts the pawl b out of gear. The lift of the pawl is limited by 
the pins at 5. As the forward motion continues, the band slips 
upon 4 ; if reverse movement begins the pawl is at once thrown 
into gear. This is used in spinning mules, also in Pouyer's 
coupling. Fig. 453, in which two pawls, each with its own de- 
vice are used. The principle involved in this device is capable 
of wide and useful application, as will be seen hereafter. 

Another form of silent ratchet is shown in Uhlhorn's coupling, 
Fig. 454. In this case the pawls b, lie close against the flanks 
of the teeth. They arc thrown into gear again by auxiliary 
ratchets, the spring pawls of which are not silent. These lift 
the pawls b, through a small angle when the engagement is 



154 



THE CONSTRUCTOR. 



completed by the self-closing action of the tooth flanks, Case 
4or 6, ? 237. 

Ratchet drills, etc., are often made with silent ratchets. Wil- 
ber's ratchet. Fig. 673, may be used for this purpose. If it is 
placed so that the axis, l, is vertical, the friction of the pawls 
against the case will lead them into gear in the forward move- 
ment and draw them out on the return movement, the friction 
in this case taking the place of any operating gear for the 
ratchets. Various other forms of silent ratchets are in use. 

I 241. 
Speciai, Forms of Ratchet Wheels. 

In spur ratchet gears the axes i and 3 of the wheel and pawl 
lie parallel to each other. These axes, however, may be placed 
in the same manner as with gear wheels so that they are 
inclined or intersect each other. A great variety of forms of 
ratchet gearing ma}' thus be made. The variations do not at 
first appear as important as they really are, but this will appear 
in the further discussion. 

A form of ratchet for inclined axes is the crown ratchet, Fig. 
675, which is used in capstans ; the wheel, a, is stationary, and 
the arm and pawl, b and c, revolve. 



Fig. 679 is a multiple ratchet of common form, with three 
pawls, in which the pawls are set a distance from each other 






Fig. 675. 



Fig. 676. 



The forms shown in Fig. 676 and Fig. 677 are for non-inter- 
secting axes, and use crown wheels also, and hence are called 
crown ratchets. 





Fig. 677. 



Fig. 678. 



By making the wheel, a, in the form of a plane wheel, and 
substituting a bolt for the pawl, some useful modifications are 
made. Fig. 678 shows a form of ratchet used on a wine press, 
in which the bolt can readily be lifted out and placed in the 
successive holes as the lever arm is moved backward aud 
forward. 

The ordinary jaw clutch coupling. Fig. 441, is really only a 
form of crown ratchet with bolt pawl. The portion on the shaft 
A is the ratchet wheel, and the part fitted to slide on the shaft 
B corresponds to the bolt b. 

I 242. 

Multiple Ratchets. 

It is frequently desired to construct ratchet gearing so that 
the minimum limit of movement shall be less than the pitch of 
the teeth on the wheel. This is accomplished by using two or 
more pawls acting at corresponding sub-divisions of the teeth. 
Such multiple ratchets exhibit a wide variety of forms and find 
many useful applications, and in many cases their true nature 
is not fully understood. 




Fig. 679. 

equal to | of the pitch. From this arrangement the wheel can 
be moved spaces equal to 

Vi, H< I. i>3. 1 73. etc-, 
of the pitch, that is, through ]< the pitch and any multiples of 
the same. This is sometimes used in saw mill feed motion, 
where a fine feed is required with a coarse pitch ratchet. 




Fig. 6S0. 

A double ratchet is used in Weston's Ratchet Brace, Fig. 6S0. 
The pawls b^ and b., are placed one above and one below the 
arm c, and act on the two parts of the double ratchet wheel 




(7i, (7,. Another ratchet drill, also by Weston, with four pawls 
is shown in Fig. 5Si. This has an internal ratchet wheel with 



THE CONSTRUCTOR. 



155 



five teeth. Double ratchets are also found in Uhlhorn's coup- 
ling, Fig. 454, and Pouyer's coupling, Fig. 453. 

If it is desired, the pitch may be halved, or divided into any 
two chosen portions, in which case the pawls ma}' be made in 
one piece, Figs. 6S2, 6S3. 





Fig. 682. 



In each of these there is one pushing and one pulling pawl 
upon the axis 3, the pitch being halved and the pawls acting 
alternately. One form shows a spur wheel, the other an in- 
ternal wheel. The form of the double pawl has caused this to 
be called an " anchor" ratchet. 

If the wheel is a so-called " face " gear, that is, with the teeth 
projecting from the face of a disc, two similar pawls may be 
used, both pushing or both pulling, and forming the same 
anchor, Figs. 6S4., 685. 





Fig. 6S4. 



Fig. 6S5. 



Fig. 686. 



If the teeth are set alternately in two concentric rings, the 
two pawls may be merged into one, as in Fig. 686. This latter 
form appears to be new. 

I 243. 

Step Ratchets. 

A very instructive form of multiple ratchet gearing is obtained 
by combining more than two pawls into one piece, and arranging 
two such pawls to work together, and this form is capable of 




Fig. 687. 

very extended application. In the ratchet combination a b c. 
Fig. 687, we have such a combination of two multiple pawls, 
with "dead" engagement, released by lifting the pawl b. The 
part a, which is impelled in the direction of the arrow is thus 
released, but is arrested again by the shoulder 2'. If the flank 



a 2' is formed in the arc of a circle from the center 3, a farther 
lifting of b will cause, without resistance, a fresh release of «, 
again arrested at /5 2", and a similar action again for the flank 
y 1'" ; the points 2, a, /3, y all lying ou a circle struck from the 
centre i. Thus a continuous lifting of b will produce three suc- 
cessive advances of a. The angle of each advance of a may be 
called the angle of advauce, and the corresponding angle of lift 
of b the angle of release. In this case the angles of advance 
are all made equal to each other, as are also the angles of re- 
lease. When the position in which 2 is arrested by the flank 
y 2'" is reached, the angle of thrust g becomes so small that 
further travel cannot well be obtained. If it is required to pro- 
vide for still further movement it can be done by making addi- 
tional teeth behind 2, as II, II', III", etc., which will engage 
successively with i at 2'". The construction of "dead" form 
of teeth is clearly shown in the diagram. As before, the angles 
of advance and release are made uniform. The mechanism as 
constructed will give nine successive engagements. The 
ratchet surfaces ou b are struck from 2, and the sliding surfaces 
on a from i ; the flanks on a with a radius 3.2'" ^ 3 }■, the flanks 
on b with a radius 1.2. 

It is to be noted that the two parts a and b are interchange- 
able in their functions, so that when the extreme notch IT'' of 
a has been reached, a ma}' be reversed in movement and b 
follow step by step to its former position. 

Such step-ratchets are seldom used in practice, but many use- 
ful applications are possible. 

In Fig. 68S is given a form of step ratchet arranged to give a 
uniform angle of advance together with uniform drop of the 
pawl. The pawl a is acted upon by the force indicated by the 
arrow, and teeth are upon a cam-shaped disc. 




Fig. 688. 

An arc with radius 1.2 passes through 3, the angles of release 
on b are 30°, and the successive angles of drop of (Z are 5°. This 
form of ratchet is used in the striking mechanism of repeating 
watches, and is known as a "snail" movement. The arm a in 
this case is frequently made ot the form shown in dotted lines 
zX A. The construction of the snail is interesting. In order to 

fulfill the given conditions the points 2.2'', 2" must lie on 

an abridged pericj'cloid ; in the given case, where 1.2 = 1.3 it is 
the form known as a homccentric pericj'cloid.* The points of 
the re-entering angles lie on a similar curve. The circles rolling 
together to describe these curves are shown in the figure T a 
rolling about i, and Tb about 3 ; their radii are inversel)' as the 
angles of drop and advance. If the parts b and a move con- 
tinuously, these circles roll on each other, for the actual move- 
ments which take place, the drops of the pawl occur as the suc- 
cessive ringed points coincide. 



* See Reuleaux's Theoretical Kinematics, § 24. 



156 



THE CONSTRUCTOR. 



In the preceding step ratchet (Fig. 6S7) the angle of drop and 
of release were given the ratio 1:2. In this case the points of 
the teeth were on cycloids, those ou a being on a pericycloid, 
those on 6 on a hypocycloid. The contact point of the gener- 
ating circle falls without the figure ou 3.1 prolonged. Since the 
radii of the circles are as i : 2 with internal contact the hypo- 
cycloid becomes an ellipse. A portion of the curve is given in 
the figure ; 3 X , and 3 Y are the semi diameters. The sim- 
plest form for the line of the teeth will be obtained by making 
1.2 = 1.3, since for this case the ellipse for one diameter of the 
base circle on b becomes the straight line 3 X. 




Fig. 689. 

If it is desired to combine in the same piece two step pawls, 
Fig. 689, of which one set shall be in tension and the other in 
compression, an anchor ratchet may be used. In this case a 
back and forth motion of the anchor permits an intermittent 
forward motion of the wheel. The anchor has ten steps and the 
wheel four teeth. This may be considered the general case of 
which Figs. 6S2 to 686 were special examples. 

Numerous interesting problems may be solved by such de- 
vices, such as the conversion of continuous rotation of one 
piece into intermittent rotation of the second. Applications are 
found in clock and watch-making. 

The various modifications which may be made in the relative ' 
positions of the axes 2.1 and 2.3 permit a very great variety of 
Step ratchets to be made. 

? 244. 

Stationary R.atchets. 




Fig. 690. 



Fig. 691. 



A stationary ratchet may be considered as a combination of a 
pair of running ratchets with the teeth facing in opposite direc- 
tions. The scheme of such a combination is shown in Fig. 690. 
From the four possible positions of the parts 2.2', II and 11' we 
may make the following double combinations : 



2 with II, 
2 with ir, 



2' with II', 
2' with II. 



The first two combinations are practically identical with the 
stationary ratchet. Fig. 691. The flanks of the two wheels give 
a notch for the space, while the teeth assume a dove-tail shape, 
and this form of stationar}' ratchet may be called a notched 
ratchet. The wheel will be firmly held by the so-called " dead " 
tooth, or when (90° — ")<;«, J 237. Many forms of this kind 
are used in practice. 




Fig. 692. 



Fig. 693 



Figs. 692 and 693 show two modifications of the notched 
ratchet. The distinction between tension and compression 

pawls disappears, since the pawl 
is the same for either action. 
If the distance between the axes 
I and 3 is made infinitely great, 
the pawl becomes a sliding 
bolt. Such a form is shown in 
Fig. 694, which is for non-inter- 
secting axes. The wheel is a 
crown wheel, and the pawl may 
have more than one notch.* 

Another form of notched 
ratchet with axes i and 3 infi- 
nitely separated is shown in 





Fig. 694. 

Fig. 695, and is in- ■ 

tended to hold a shaft 

from longitudinal 

motion, being used Fig. 695. 

in connection with 

the disengaging gear of hoisting machinerjf, lathes and other 

similar machines. 

In this case the radius a is infinitely great ; the wheel a be- 
comes a shaft. 

The combination 2 with II' and 2' with II of Fig. 690, if we 
make 3.2^ III . II, gives a stationary ratchet of the form 
shown in Fig. 696. 




Fig. 697. 



The pawl becomes a segment of a cylinder and works always 
in compression, or in the modification given in Fig. 697, always 
in tension. This form may be called a cylinder ratchet. The 
form of Fig. 696 has many applications, as, for example, the 
Thomas' Calculating Machine and similar work. 



* This form of ratchet will be recognized as similar to the common jaw 
coupling. The shaft A carries the crown wheel a, the bolt corresponds to 




the other half of the coupling d. The shaft H carries the part d. the latter 
sliding upon a feather. 



THE CONSTRUCTOR. 



157 



The cylinder b may be entirely cut through as in Fig. 69S, so 
that the segment shall fall entirel3' within the surrounding 

circle. When it is placed op- 
posite the teeth the wheel may 
be revolved in either direction 
as far as desired. If this move- 
ment is to be limited, as, for 
example, to a given pitch, it 
can be accomplished by cut- 
ting a corresponding space in 
the cjdinder, such as is shown 
in Fig. 699 a. 

It is not necessary that the 
spaces in the wheel a should 
conform to the circular profile 
of the cylinder b (see \ 237) ; 
the thrust is at two points on 
the right and left of i . 3, and 
it may be formed as at b, or 
pin teeth used as at c. This 
last figure shows the modifi- 
cation made in the notch of 
Fig. 698 to reduce the back- 
lash of the wheel a. In Fig. 699 a the pitch circle of the pin 
gear a passes through the axis 3, and the gap in the cylinder is 
increased proportionally. When the wheel is impelled in the 
direction of the arrow, the pin 2 slips into the space in the 




Fig. 698. 




Fig. 699. 

cylinder as soon as the opening is turned towards it far enough, 
but cannot pass out until the cjdinder has turned back the same 
distance in the opposite direction, thus forming an intermittent 
pitch movement. 

This idea is more fully carried out in Fig. 699 e. In this case 
the inner profile of the space is concentric with the outside of 
the cylinder, as was also the case with the form shown in Fig. 
697. In this case the tension and compression pawls are practi- 
cally combined in one. When the opening moves into the 
proper position, the pin 2 moves to the point 2', and completes 
the remainder of the pitch movement when the C3'liuder moves 
to the left again. This form may be made free from backlash 
by making the outside of the cylinder fill the space between two 
teeth, as in Fig. 700. If it is required that the intermittent 
movement should divide the pitch into two equal parts, the arc 
of the pitch circle of <7, which is the measure of the thickness 
of the teeth, must be equal to the arc cut off by the space in the 
cylinder. If backlash is permissible, the thickness of tooth 
may be reduced.* 




Fig. 700. 

If we compare the various forms of cylinder ratchets mth the 
notched ratchets, as, for example, in Fig. 692, it will be seen 
how the one may be derived from the other. If the pawl of 
Fig. 692 is given a row of teeth similar to the tooth 2, placed in 
a circle about a centre 3, and a space cut in a of the circular 
profile indicated, we obtain the same general and important 
form as is shown in Fig. 698. 

In a .similar manner tie notched ratchet can be derived from 
the cylinder ratchet, and also inverted by transposing the parts, 



* If the preceding forms are compared with Fig 6S2, a similarity will be 
noticed. The " dead" ratchet with pawls of circular profile, of Fig. 6S2, are 
here, in Fig. 699, replaced by a gap of small angle ; the compression pawl is 
at 2, the tension pawl at 2', the arc 2 — 2' is made very small, and the re>a- 
tive diameters very dififerent. 



and all the modified forms obtained. The interchangeability 
of the two parts gives the midway form shown in Fig. 701, in 
which both pieces are the same, each being wheel and pawl for 
the other.* 




Fig. 701. 

For the varied positions which may be given to the axes, a 
wide variety of cylinder ratchets can be made, many of these 
possessing useful applications. If the axes are at right angles, 
the cylinder may become a disc, as iu Fig. 702 ; this form being 
used in Thomas' Calculating Machine, in which case the wheel 
a is made with but a single tooth. 




Fig. 702. 



Fig. 703. 



Fig. 704. 



The form shown in Fig. 703 is derived from the globoid gear- 
ing of Class III, \ 224, the ratchet being a cylindrical notched 
ring. Fig. 704 shows how a pitch ratchet can be made on this 
principle. 

An examination of the preceding forms of stationary ratchets, 
in which the pawl consists of a revolving member with a gap 
cut in it, will show one common property in all of them. This 
is the fact that an intermittent motion produced by successive 
release and engagement may be made either by a continuous 
rotation of the cylinder or by an oscillating movement. If, 
therefore, we have a continuously revolving shaft to deal with, 
or a vibrating member, the desired release or intermittent ac- 
tion of the part to be acted upon may in either case be ob- 
tained. Both forms are found successfully applied in actual 
practice. 

?245. 

R.^TCHETS OF Precision. 

If we imagine the running ratchet of Fig. 682 so modified 
that upon the release of the pawl 2 that at 2' shall enter at a 
point nearer the tooth than the middle of the pitch, as there 
shown, the principle will not 
be changed. If this modification 
is made to such an extent that 
the angle & in both cases be- 
comes zero, ;'. <?., the pawls so 
made that one enters into eu- 
gagement at the instant of re- 
lease of the other, we have the 
form shown in Fig. 705. 

In this case the wheel a, be- 
ing impelled in the direction — 
of the arrow, can pas6 the 
points of both pawls at once. 
The slightest movement of the 
member b in either direction, 
however, will bring either 2 or 
■2^ into »ogagement and hold 
the wheel. This form is called 
a Ratchet of Precision, the 
especial one given being a 
running ratchet. 

The principle is capable of various applications, and is also 
suitable for stationary ratchets, two forms of which are shown 




Fig. 705. 



* This form is sircfilar to the rauning ratchet of Fig. 671. 



158 



THE CONSTRUCTOR. 



in Figs. 706 and 707. In the latter case the pawl assumes the 
form of a bolt, shown in the illustration with several notches. 





ViG. 706. 



Fig. 707. 



The practical applications of ratchets of precision are numer- 
ous, and examples will be given hereafter. 

? 246. 

General Form of Toothed Ratchets. 

We have alread}' seen that several forms of ratchet mechanism 
which have been described possess numerous points of similar- 
it}', and may be reversed and derived from each other, and 
hence it is not unreasonable to expect that some general foim 
may exist from which the various special modifications can be 
derived, and in which the distinction between ratchet wheel 
and pawl, or checked and checking member, shall not exist, 
but each shall appear in both. 




Fig. 70S. 

This general form is actually found in the combination of two 
disc face wheels (§ 211), with their centres carried on the same 
bar, Fig, 70S, in such a manner that the teeth of both shall 
engage and be engaged by the other. In the illustration is 
shown such an arrangement made for a stationary notched 
ratchet. The wheel b engages as a pawl with the wheel « at 2 
and 2', and if it revolves a space of one-half a pitch, a is re- 
leased. If a, however, revolves any given odd number of half- 
pitch angles only, b will be checked, and a become the pawl. 
In both cases we have a ratchet of precision of the same type 
as in Fig. 706. 

The pitch ratchet with anchor pawl may also be thus derived ; 
it is true the anchor form cannot so readilj' be shown as a pair 
of similar wheels, but it is clearly only another form of the 
same problem. The zig-zag ratchet, notched ratchet, step 
ratchet, or their combinations are all reducible to this general 
form, the only condition being that the direction of the force in 
the position of engagement of the checking member shall be 
such that the checked member cannot revolve. The intermedi- 
ate forms show the "pawl lifting" action, 'i 237. It is evident 
that in some cases the checked member maj' have a forward 
movement, and in others a reverse movement. Since here, as 
in \ 255, we may consider the link ;: as a checked member when 
the wheel is held fast, we may, from the combination of these 
parts, obtain four kinds of ratchets, viz. : 

1. c, stationary ; a, checked ; b, checking. 

2. c, " b, " a, " 

3. (7, " c, " b, " 

4. b, " c, " a. 

As a general statement of the fundamental principle we have : 
A toothed ratchet consists 0/ a combination of a pair of gear 
lulieels, or of portions of gear wheels, in which the teeth are so 
made that for certain positions of the wheels the resultant of the 
pressures on the teeth of one of the wheels either passes through 
lis axis, or differs from such direction by less than the angle of 
frictio7i. 



Dimensions of Parts of Ratchet Gearing. 

The great variety of ratchet gears in use makes it almost im- 
practicable to prepare any compact rules for the determination 
of the dimensions of the various parts. The general proportions 
can be obtained for the various forms by comparison with simi- 
lar preceding devices. For spur ratchet wheels similar propor- 
tions may be used as for spur gears with thumb-shaped teeth. 
\. 212. The action of the pawl tends to produce shocks and this 
must not be overlooked in determining the thickness of the 
teeth. It is generally most convenient to give the pawd a curved 
profile, in which case the discussion of combined resistance, § 18, 
is to be considered. Pawls which are subject to frequent vibra- 
tion are best made of steel, as are also those in whicli the super- 
ficial pressure is high. 

I 24S. 

Running Friction Ratchets. 

The mechanical devices which are constructed to modify the 
relations between two moving bodies by means of friction, may 
be called by the general term of friction clutches* Such a de- 
vice, when so arranged that one member opposes a positive 
frictional resistance or check to the motion of the other in one 
direction under the action of an impelling force, constitutes a 
friction ratchet. Such de- ( 

vices may be divided, as j_,. 

before, into running and 
stationary ratchets, ? 235, 

and the first form will now '^^1 

be considered. 

lu Fig. 709 is shown a 
friction ratchet for parallel --c\fC 

axes. In this case the fric- -'-'^-'Cl 

tion block b is carried by g^-"'"" / 

the friction with the wheel \ 
a, when the latter begins '. 2> 

to revolve in the direction \ 
of the arrow, that the pawl \ 

link c is crowded against 
the axis 4. The radial com- 
ponent O, in the direction 
4 . 3, exerts a pressure upon 
the brake block b. We 
also have the tangential 
component S, which we 
may consider as composed 
of two forces S-^ and S,, act- 
ing in the same direction, 
which hold the friction at 
I and 2 in equilibrium. At 3 we have two opposite forces S^ 
and S^ which are capable of resisting the friction at 3 and 4 res- 
pectivel}'. 

The moment 31, of the four friction forces is : y!/= [S-^ + S^ 
— Sj — 5,) (a-\- b). If we give the angles the symbols shown in 
the illustration, and make i .2 = (r, 2.3^15, 3.4 = c, 4. i =rrf, 
and call the radii of the several journals «j, b^, and c^, we have : 




Fig. 709. 



5,: 



^Qfa, 
a + b' 

Qf , 



5,= 



Q_f_a 
(7 4- 4' 



S.= -^f-^^ 



and S, 



5, = --!-(«'« + rfy) 



dy 



But we also have (a -j- b) sin a = c sin y. 
From this we get 

c cos a cos (7 c cos a 

(a + b) cos a [a -\- b) cos a 

This gives for 31 : 



A_rf 



s., = - 



M=Qf 



^) 



{a + b) 



a -\r a-i i b-^d 

a-\- b cos- a c{a -\- b) cos a 

The force P which acts at 2, to revolve the wheel in the direc- 
tion of the arrow, may be considered as a couple. We then 
have for 3/^ Pa : 

Pa ^ , r'? + "1 



a + b 



= Qf 



[ 



a + b 



DS*' C \C 



__'V'_ L f: 

[a + b)coza c 



m 



* This term only partially expresses the general scope of the German word 
*' Bremswerke," \ox which there is no exact equivalent in, English. — Trans. 



THE CONSTRUCTOR. 



IS9 



Pa 
But (9 is a function of/', aud iu fact we have — ,— 7 ^(J tau a.* 
~ a -\- 



This gives : 



sin a cos n — y sin a 






\.\ 






b^d 



b) cos a 



+ ^ 



?)] 



c [a ■ 

and since the angles cr and a are small, and become smaller 
under the action of the pressure, a sufficiently close approxima- 
tion will be obtained by putting : 



s/K^-G 



^)] 



+ b \^c(a + b) 

The following conditions must be noted. If an independent 
force outside of Q exerts a normal pressure IV upon the circum- 
ference of the wheel, the friction iV/will diminish the force 
acting to turn the wheel backward. If this is to enter into the 
resistance which is produced by Q, the magnitude of a as given 
by equation (233) must be modified. If jV becomes sufficiently 
great, Q maj' become zero ; in such a case we obtain a stationary 
instead of a running ratchet. 

The pressure 7? on the pawl may become very great. We 

have J? = — ^ — which may be made approximately : 



J? = 



P_a 

{a -+- 6) sin o- 



(234) 



Mjcantple. — I,et a -- 
/"at all four points = 



T4.2", a-i = 1.6", d = 2", ^i = 0.6", c = ir.8", ^1 = 6", and 
o.io,-f we have d =^ a -i- ^ -i- c = 2S" approxiinatel}', and 

15 s 0.6 -I- 28 0-6 ^ 



sin 5 ^ o.io ( — — 

\x6. - 



+ 



^8 0-6 \ 



whence sin a- < 0.0834, which gives a- = 4° 47' 
make a- = 5%°, or sin o- = 0.07S7, and then get R 



To be on the safe side xve will 
- = II. 17/*. 



16.2 -j- D.0787 
The exact length of d will be very slightly' less than a + i + c. 

As will be seen the ratio comes out unfavorably. The method 
of remed}'ing this will be discussed hereafter. 

The pawl c may also extend within the circle of the wheel, as 
in Fig. 710, in which a is an obtuse angle. The axis of the pawl 





Fig. 



710. Fig. 711. 

may be either at 4 or 4', on 3 . 4 prolonged ; the pawl is in this 
case a tension pawl. If a is made an internal wheel, we have 
the arrangement shown in Fig. 711, the pawl being under com- 
pression. 

Especially noteworthy are those cases in which one or more 
of the axes are infinitely distant. In Fig. 7 [ 2 is shown the case 





Fig. 712. 



Fig. 713. 



in which the length of pawl and also d and ^, are of infinite 
length. We have for the angle of thrust, from (233) 



P 



^- The moment of the frictions produced at 2 and i b}' the force Pis 
{a + ai) — Pill = P^- 

t If various coefficients of friction are to be used we have for 5i, S., Ss and 
S4, corresponding values yi, /n, J3 and 74. 



c- 



Ho-] 



(a + b) [a + , 

When rt, is very small, release is difficult, and the arrange- 
ment does not appear to be verj- practical. If the arm a is made 
infinitely long, so must also a^, aud we get the case of Fig. 713. 
The wheel becomes a sliding bar. The relations 



o</ 



[-<^)-(v)] 



give excellent action. 

If with a and ^i w"e make 
construction of Fig. 714. 



c infinitely long, we obtain the 



1^ 




F19. 714. 




1 2 



Fig 



d 



Fig. 716. 



The conditions give : sin u ^/{2 — i). The joint at 3 insures 
full bearing for the surfaces at i, 2 and 4. This is also the case 
with the joint at 4 in Fig. 715, iu which a, a^, b„ c and (/are in- 
finitely great, while b is the difference between two infinitely 
long but opposite distances, and hence is finite. We have the 
relation c ^y (2 — i). By omitting the joint we obtain the 
simple construction shown in Fig. 716. The friction block is 
in the form of a ke_v or wedge, as in Fig. 715, aud the number 
of parts reduced to three (see also the following section). 

The results as determined by calculation are not always prac- 
ticable for the desired ratchet construction, which shows that 
the selection of the relation between the parts must be made 
with judgment, and care taken that those pieces which are sub- 
jected to heavy pressure shall not readily be deformed. As the 
preceding example indicates, the small size of the angle a ren- 
ders it an important point for consideration. In this case the 
actual length oi d is oul}' about ^j/' greater than a-\-b-^ c. The 
pressures Q and R act to lengthen d and shorten a, b and c, aud 
if /'= 100 pounds the difi'erence may readily = jV", so that 
with only jV' additional wear, a becomes zero, the parts b, c 
pass the centre, and the ratchet action ceases. This will indi- 
cate the conditions under which the ratchet becomes a useful 
device. 

The numerical magnitude of the parts flj, c and d can be 
chosen so as to render the unavoidable wear least hurtful. This 
may be done with the sliding ratchet, Fig. 713, b}' making the 
length c sufficientl5' great. It is also important to devise means 
to prevent the block and pawl from being forced past the 
centre. A method of accomplishing this is to substitute for the 
pin joint between the block and pawl, a curve or cam bearing, 
as in Fig. 717. 




Fig. 717. 

If the block is given a circular profile from the centre i, and 
the pawl an evolute outline developed from a circle about 4, 
with radius d sin 0, we shall have cr nearly constant, notwith- 
standing the elastic yielding of the parts and the unavoidable 
wear. If the yieldiug between i . 4 is great, the radius of the 
circle on which the evolute is generated may be made somewhat 
greater than d sin a. This construction appears to be new. A 
number of similar modifications may be made, for which see 
the following section. 



i6o 



THE CONSTRUCTOR. 



It is desirable to examine the value of the coefficient of fric- 
tion/, in order to increase it at the point 2. This cannot well 
be done by choice of material, since wood can scarcely be used 
m many cases, and lubrication of the rubbing surfaces is essen- 
tial. The application of wedge profiles to 
' wheel and friction block enables greater fric- 

tion to be obtained, Fig. 71S, as in the case of 
wedge friction wheels, ^ 196. Instead of the 

f 



.._./ 



f>- 



coefficient/ we have the value 



If 



the wedge angle 6 = 60° this gives 2/; for 9 
=: 30°, nearly 4/. By combining this prin- 
ciple with the preceding forms, some very 
useful devices may be made. 

I ._, It is desirable to arrange the application of 

! ' ; the force R so as to exert as small a distorting 

'; i i action upon the parts as possible. This may 

i 0j' sometimes be done by arranging two or more 

\\ ; friction pawls of similar kind to act upon one 

i ' wheel. Some examples of such devices will 

■!■ be found in the following section. It must 

';■; not be forgotten that the conditions for 6 are 

s not changed by the repetition of parts, since 

1 the numerical value of /'does not enter into 

Fig. 718. its determination. 

There is yet another form of friction ratchet 
which is capsble of being made very useful. By an examina- 
tion of formula (223) it will be seen that the influence of the di- 
mension (2, is almost as great as that of a itself. If we inlrrease 
«! to nearly the same magnitude as a. Fig. 719, we may approach 





Fig. 719. 



Fig. 720. 



closely the minimum value of a. This carries with it the dis- 
advantage that the frictioual resistance to the backward and 
forward movement at I, is greatly increased, but this effect may 
be avoided by making a special bearing for the friction block 

^ axis and rearranging the 
parts somewhat as shown 
in Fig. 720. The attempt 
of a to move backward 
causes the pieces b and d 
to press upon the rim of 
a from without and with- 
in and grasp it firml5'. 
The angle c may now be 
made twice as great as in 
the previous forms with- 
out danger, all other 
things remaining the 
same. A practical form 
of this device is shown in 
Fig. 72 1 , as applied to saw 
mill feed motion. Here 
the screw motion F G \s 
intended to permit of a 
suitable degree of play 
for the lever <:.* If we 
make Sj > n, we have the 
form shown in Fig. 722, which seems qviite practical, and when 
applied to a friction rack we obtain the form in Fig. 724. We 
shall return to the consideration of these double friction ratchets 
hereafter. 

It must be remembered that these forms of friction ratchets 
are also applicable to other positions of axes and scune resulting 
devices are in practical use. 




Fig. 721. 




Fig. 722. 



Fig. 723. 



? 249. 



Running Friction Ratchets. 

If the force to be transmitted is not very great, the intermed- 
iate friction block may be dispensed with and the curved con- 
tact surface be made directly upon the pawl. This reduces the 
mechanism to three parts : the wheel «, pawl b, and arm or con- 
necting bar c. Fig. 724. 




Fig. 724. 

This form may be called a clamp ratchet, or since the pawl 
resembles the thumb-shaped teeth already described, the term 
"thumb-ratchet " may be used. The determination of the angle 
B may readily be determined by what has preceded, and the fol- 
lowing relations established : 






(235) 



A suitable profile for the thumb pawl may be obtained as in 
Fig- 717) tiy using the evolute upon a base circle of radius c sin 
a, f bout 3 as a centre. This may be approximated by a circular 
arc struck from M, in which 3 M and i M are at right angles to 
each other. 

If a and c are made infinitely great we have a form similar to 
Fig- 7I3> the straight profile 2 being an evolute of infinitely 
long radius, and the profile 3 a portion of the circumference of 
an infinitely great cylinder. 

If the wheel be made a wedge friction wheel we have the 
form shown in Fig. 725. The wheel may be made with several 
grooves, by which means the pressurs on each surface can be 
reduced (see \ 196). 




Fig. 725. 



Fig. 726. 



A variety of modifications can be made in the arrangement of 
the pawls. A clamp ratchet in which a repetition of pawls is 
used to distribute the pressure, is Dobo's ratchet, Fig. 726, which 
is very effectively used by A. Clair in his indicator. f 

If we adapt the idea of Fig. 717 to a revolving journal using 
the "thumb " pawl at 3, we obtain a very useful modification of 
the clamp ratchet. The curve which is applied between 2 and 
3 may be variously arranged. A very simple form is obtained 



♦See Goodeve. Elements of MecUauism, London, 1S60, p. 49. 



fSee Morin, Notions geometriques sur les mouvements, Paris, 1861, p. 200. 



THE CONSTRUCTOR. 



i6i 



by making the curves at 2 and 3 portions of the same circle, and 
the corresponding curve at 3 so found as to produce the required 

clamping action. 
The clamping piece 
b becomes a cylin- 
der, Fig. 727. 

If we make the 
angle O 2 3 = rf, 
prolong the radius 
3 O to N, then -will 
3 O yVbe the normal 
to the curve at the 
point of contact 
with b at 3, since 
the angle 3 . 3 (9 = 
S. The curve for c 
is an arc of a circle 
struck from a centre 
DI, on 3 (9 N, found 
by making i jl/per- 
pendicular to 3 O N. This curve is practically correct for a 
smaller clamping C3'linder as at O' 3', since the angle of thrust 
is very nearly the same as at (9 i . 2, or in other words the ef- 
fectiveness of the clamping action is not impaired as the cylin- 
der is reduced by wear. 

T 




The pressures at 2 and 3, Fig. 72S, are T = , R 



cos cr 



7—, whence O - 

cos- IT 



P 



(■ 



(a + a,)f 





Fig. 728. 

A practical application of the preceding form is shown in the 
checking device for sewing machines. Fig. 729. In this case a 
ball of rubber is substituted for the cylinder. Another similar 
device is the ratchet check used on the old LangenGas Engine, 
Fig- 73°- III this case a number of roller checks are used in 
order to distribute around the wheel a. 

The whole forms a sort of continuous ratchet gearing in which 
the backward and forward movement of c imparts a continuous 





Fig. 730. 



Fig. 731. 



forward movement to the wheel a. AVhen c moves in the direc- 
tion of the arrow II, a is clamped and driven, while the parts 
are released when the motion is reversed in the direction I. The 
action of the centrifugal force tends to keep the checking cylin- 
ders in contact with the outer ring, and so insure prompt action 
upon the reversal of motion. The piessure upon these roller 
checks in the Langen Gas Engine was very great ; wrought iron 
rollers wore out rapidly and phosphor bronze was substituted, 
although even these gradually altered their form under the 
pressure. 

Another ratchet check used by Langen for the same purpose, 
is shown in Fig. 731. Here again we have a repetition of the 
parts, and also a return to the friction block, the rollers occu- 
pying the place of pawls. Comparing this with Fig. 709, the 
curved bearing surfaces correspond to the journals 3 and 4, and 
the action is similar to Fig. 727. The block b is arranged so 
that full clamping is obtained in a quarter turn. Friction ratch- 



ets with double 
clamps are also 
used as in Fig. 721 
and the same 
principle appears 
in Fig. 732, which 
shows Saladin's 
"friction pawl."* 

A similar de- 
vice is shown in 
Fi.g- 733. as ap- 
plied to a rod 
movement, and 
upon inspection 
the resemblance 
to the action of 
the "thumb" 
pawl will be seen.f 

As long ago as 

1798 Hornblower applied this idea to a rotary engine as shown, 
in Fig. 734.; 






Fig. 733- 



Fig. 734. 



250. 



The Rei<ease of Frictiox Pawls. 

The release of a friction pawl under pressure requires a cer- 
tain degree of force, since there is always a friction between the 
rubbing surfaces which is at least equal to P, which must be 
overcome if the pawl is to be released under pressure. The 
release is to be effected under 
quite different conditions 
from those which obtain with 
toothed ratchets in which, 
for example, with "dead" 
engagement, only the "y"th " 
part of P is exerted at the 
pawl point. The force re- 
quired for release may be 
somewhat reduced by combi- 
ning the action of two sets of 
friction surfaces of opposite 
direction of engagement. 
Figs. 735 and 736. The mo- 
tion in the direction of the 
arrow tends to draw the pawl 
2 into closer engagement, 
and at the same tiiiie to release that at 2'. By altering the 
relations of the distances 4-3 and 4-3', etc., any proportion of 
the moment may be used to hold the parts in gear. These 
forms appear to be new, and may be called "throttle ratchets." ?; 

? 251. 

Stationary Friction Ratchets. 

A stationary friction ratchet may be defined as one in which 
the clamping action is not dependent upon the direction of 





Fig. 735- 



Fig. 736. 



* See Bulletin vou MiilhauseB, XII. 1S3S, p. 296, also Salzenburg's Maschine 
details. 

t A similar arrangement will be found to exist in the ring spinning frame. 
Here the pawl b, Fig. 732, is made of wire and held at 4 by the thread pass- 
ing through an eye. Since the angle o- is made greater than we have taken 
it above, a tension or brake is necessary. 

X A similar device is used by Carter, both being found in Farey's Steam 
Engine, PI. XV, Figs. 8 and 9, also in Severin's .\bhandlung, p. 141. 

3 It may be noted that friction pawl actions are found in nature. Some 
fishes have such connections to certain bones or spines which they can 
thus elevate or depress. See O. Thiele, Die Sperrgeleuke eiuiger Welse, 
Dorpat, 1879. 



3 62 



THE CONSTRUCTOR. 



rotation of the wheel. Such a ratchet is shown in Fig. 737. 

I and 4 are parallel axes, the block acts with a radial pressure 

Q, such that for the 
circumferential force 
dz /'the following con- 
ditions may exist : 




trigger is that part of the pawl upon which journal 5 of the releasing cam 
is carried. The tooth profile at 2 should be " dead," but this is not the case, 
as the curve is struck from the centre 3. The bearing points on pawl and 
sector are of steel, separately inserted. The force which closes the valve is 



Qf[a^ fli) >P«,or: 



If Q is less than 
the right hand expres- 
sion, /^ will only be par- 
tially opposed, there 
will be motion from a 
toward d, with slip- 
ping at 2, or in other words, we have a brake, see \ 248. 
This construction is frequently applied, although it requires a 
relativelj' large force at Q' , acting through the lever c c', giving 
increased pressure on the axle and much wear on the block. 
Various forms oi lever connection are used to modify the ratio 

O' : O- By clearing the angle 
which the axes l and 4 make 
with each other, variotis con- 
venient modifications may be 
made. The general scheme 
of such constructions is indi- 
cated in Fig. 73S, in which the 
toggle connection gives a high 
ratio of Q' to Q; the block 
being guided in slides. By 
making a an internal wheel, 
a very practical arrangement 
is obtained as shown in Fos- 
sey's coupling, Fig. 450. 
Koechlin's coupling. Fig. 449, 
is also another form of fric- 
tion ratchet gearing, the pres- 
sure in this case being applied 
by the medium of a right and 
left hand screw. The same is 
true of other forms of friction 
coupling, and the various me- 
thods of applying the pressure ■ 
and reducing the wear, given 
in ? 248, may also be applied 
in the design of mechanism 
for the purpose. 




Fig. 73S. 



^252. 

REI,E--^SING R.4.TCHETS. 

Following the classification given in J 235, we have first dis- 
cussed the various forms of ratchets for the general meaning of 
the term, and the five special classes remain to be considered, 
the next being the so-called Releasing Ratchets. Such ratchets 
must be considered primarily with regard to the question of re- 
lease. When the release is to be effected by hand, various 
forms of handles or other connections to the pawls are readily 
devised. In most cases, however, the release is automatically 
effected, in which event, some mechanical tripping device is 
required. 

The resisting force in such gearing is practically the same as 
the force required for release. It is applied usually bj' weights, 
springs, steam or air pressure, etc., and is variously intended to 
cause the released member to act with a predetermined velocity, 
either slow or rapid, as may be required. Many millions of releas- 
ing ratchetshave been made for gun locks, and the various forms 
of releasing valve gears for steam engines, introduced by Corliss, 
but first invented by Sickles,* are of this class. In designing 
releasing valve gears, it is important that the valves should be 
closed quickly yet without sudden shock, and hence some form 
of buffer is essential. It is in the various devices for applying the 
force, for releasing, and for cushioning the released force that 
the many gears differ from each other. The original form of 
Corliss valve gear, and the modified form of Spencer & Inglis, 
are but little used on the continent, but these are well known, 
and hence examples will be given of some of the numerous 
modified trip valve gears which have been put into practical use. 

Exa}iiple I. — Valve gear by Cail & Co., Paris, Fig. 739. a is the driven 
piece, a sector with one tooth, fast to the valve stem : b is the pawl ; c the 
arm, loose on the hub of a ; b' is the pawl spring; d the releasing cam. The 




Fig. 739. 

exerted by a spiral spring acting on the rod/, and the valve is opened by the 
rod connecting the arm c^ with the engine motion. The cushion is effected 
by an air dash pot. also acting through the rod /, and the instant of release 
is determined by the governor. 
Example p.— Valve Gear by Wannich, of Briiun. In this case there are two 




vry 



* F. E- Sickles, of Providence, R. I., took out his first patent for a " trip 
cut-off" valve gear in 1842. 



Fig. 740. 



flat slide valves to be operated by the reciprocating movement of the piece c. 
It will be seen that this is a form of ratchet rack ^rearing. The valves are 



THE CONSTRUCTOR. 



163 



closed by steam pressure acting upon small auxiliary steam cylinders on the 
rods a, tJie cushion being provided by air buffers as in the preceding exam- 
ple. There are double pawls (^, i^, with "dead"' tooth profiles, faced with 
steel ; c is the pawl carrier, moved back and forth by the rod c' ; b' b' are the 
triggers, and ^rfthe releasing stops, the latter shown in three successive 
positions ; e is a guide rod. The rod c receives motion from an eccentric ou 
the engine shaft. 

Example 3. — Valve Gear by Powel, of Rouen, Fig. 741.* This is a form of 
rod ratchet gearing with bolt pawl. Here b is the driven piece in which the 




Fig. 741. 

bolt h and its spring are carried. The rod a is moved up and down by an 
eccentric. The piece c is guided at Cn. The trigger d acts sooner or later, as 
the governor changes the position of the trip £. The force to close the valve 
is steam pressure acting on the upper part of the rod Ci, which also carries 
an air buffer. 

The use of releasing ratchets in valve gear of steam engines 
is very old, being found in the old Newcomen pumping engines, 
and in the Cornish engine a similar gear is used to-day, while 
in recent times trip valve gearing of various designs have come 
into extended use, and some of the forms are shown in Figs. 
670 and 671, not only for closing the valves, but also for open- 
ing them. These latter valve gears are intended to be operated 
by direct connection with the piston movement, while those in 
the preceding examples are operated from revolving crank 
shafts. 

Releasing gears which are to be operated by reciprocating 
members, are sometimes constructed on quite a different prin- 
ciple, viz. : that of a weighted lever in nearly unstable equi- 
librium, so that it can be caused to fall to the right or left by 
means of a slight thrust, and so operate a releasing member. A 
form which was formerly much used, in which the lever is 
carried on a horizontal axis, is shown in Fig. 742. 





Fig. 742. 



Fig. 743- 



When the weight G is in the vertical position i . 2, the pres- 
sure acts directly downward upon the axis, the journal friction 
acting as a ratchet. The form is sometimes used on planing 
machines, screw-cutting machines, etc. 

Another form is shown in Fig. 743. Here the pressure is due 
to a spring, acting through a link 3-2 upon 2 . i. 



A third form is that used in Shanks' planing machine, Fig. 744. 
In this case the lever, with 
its axis a, is at right angles 
to b, and the latter is pro- 
vided with a roller. The 
limit of measurement of « 
is between 2' and 2". 

The forms of tumbling 
ratchets described in \ 239, 
may be adapted as releas- 
ing gears, but it must not be 
forgotten that in such me- 
chanisms provision must 
be made for the middle 
position of the ratchet. 

A fourth form of tumbling gear, of which, indeed, there are 
many varieties, is the so-called "loop" of Hofmann's valve 
gear. Fig. 745. The loop a is made in the arc of a circle from a 





c' ^ 


a 








■J 


.1 

1 


"^ ^k. ^ 




-(<!*- 

*! 


^. Li,-^i.vl.4 J 


i 


""\<C\f-yr J 



Fig. 





'>^ 



Fig. 745. Fig. 746. 

centre at 2, 5 is a heavy roller, with additional weight suspended 
at d' . When the loop or curved link is in either of the positions, 
3o or 3', the weight acts to continue the motion in the direction 
in which it started until the limit of travel is reached. 

A swinging arm b may be substituted for the slot and roller, 
Fig. 746, and it will be seen that during the movement from the 
position 2;j 3g to 2' 3' the tumbling action will take place and the 
arm « be carried over. The similarity to the previous tumbling 
gear will be apparent. If 2. 3 bemadeinfinitely long, the loop will 
become straight and the two forms will coincide. Hofmanu has 
made the analogy to a ratchet train more complete by placing a 
ratchet so as to engage with the point 3 in the positions 3,, and 3', 
the release beingmade at the proper time by means of a cataract.* 

In some cases it is desirable to make a gearing which shall be 
released by the action of a very small force. For this purpose 
a second releasing gear may be introduced, itself being readily 
released, and by its action permitting a blow to fall upon the 
trigger of the main gear. Such a device forms a releasing gear 
of the second order. Such an example is shown in the hair- 
trigger of a rifle. t Releasing ratchet gearings of higher orders 
are also found in textile machinery, as in the Jacquard loom, 
also in the striking gear of tower clocks and of repeating watches. 
Another example is found in the relay of the Morse telegraph, be- 
sides many other applications which will be considered hereafter. 

? 253. 
Checking Ratchets. 

Checking ratchets are used in a great variety of machines, but 
their principal applications are found 
in machinery for hoisting and lower- 
ing heavy loads, as in mine lifts, ele- 
vators, and the like, to guard against 
accidents in case of the breakage of 
the 2 ropes. In the opinion of the 
writer these devices have not been as 
yet regarded as they should be, 
merely special cases of ratchet con- 
struction, and as such capable of util- 
izing all the various principles here- 
tofore considered. When examined 
in this light their study will be greatly 
facilitated. 

As a scheme of a general system for 
checking ratchets a rod friction ratchet 
may serve. Fig. 747, in which the rod 
a is held stationary, the loaded mem- 
ber d carries the ratchet, and the pawl 
c and friction block b are held out of 




* Further details of this and the preceding gear will be found in the Aus- 
trian Report on the E-xposition of 187S. Section ou Steam Engines by A. 
Riedler, Vienna, 1S79. 



* See Zeitschrift des Vereins deutscher Ingenienre, i860, Vol. IV., p. 209. 
t Such hair-triggers were ingeniously applied in former times upon cross- 
bows. 



164 



THE CONSTRUCTOR. 



engagement by the releasing lever f, and rod e as long as the 
hoisting connections^- and /;, are under stress. If the tension is 
released, the ratchet is thrown into gear and the parts clamped. 
If a toothed ratchet is used instead of a friction device, the 
block b is omitted. According to the manner in which the 
various constructive details from a to h are arranged, we obtain 
the various systems of checking ratchets which have found 
practical application. 

A collection of such devices was exhibited by the Industrial 
Association (Verein fiir Gewerbfleiss) in 1879* More than So 
designs were shown, of which only a few can be described. 
Many of the device? were rather designs for improved construc- 
ti ^n as regards strength and rigidity, rather than examples of 
im. chanical ingenuity. 

1 1 most cases the clamping action takes place upon the up- 
rignt timbers of the shaft ; sometimes guide ropes are used. 
The greater number of designs shown used friction clamps, 
those of the type of Fig. 724 being shown, the thumb pawl 
being roughened, however, or finelj' toothed. The one which 
showed the most evidence of careful constructive design in 
accordance with the principles previously laid down in 'i 24S, 
was that of Hoppe, shown, as attached to each side of the 
hoisting car, in Fig. 74S. 




Fig. 748. 



The form of friction pawl used is similar to that shown in 
Fig. 713, there being four pawls on each side of the car, or eight 
in all. The clamping action takes place upon the guide bars (7, 
made of T iron, as shown. At i are the guide rods between b 
and a ; at 2, the double clamp blocks of hardened steel, which 
are connected at 5 to the coupling rods e, e. The actuating 
springy is a torsion spring (see Fig. VII, p. 19 ; also Fig. VIII, 
p. 19), secured to the roof of the car at^, g, and operated by 
the releasing gear^ at 8, and transmitting action from 6 to 5 by 
the rods e, e, the connection being made by the links 9 to the 
double chain in such a manner that the army" cannot be drawn 
too far out of position. The proper adjustment of the pawl 
arms is obtained by the keys on the rods c, c. Hoppe has taken 
into consideration the fact that the angle j, see (233), must not 
pass beyond certain limits, or too great pressure would be ex- 
erted on the frame d, -d, and hence has provided stops in the 
frame for the tiavel of the pawls c, c. The parts are so propor- 
tioned that a load of double that ever placed upon the car would 
be supported by the friction clamps before there would be an 
appreciable elastic yielding of the frame. The adjustments of 
the rods c, c provide for the change of relations due to wear. 
This apparatus does not bring the lowering car to a sudden 
standstill in case of breakage of the hoisting gear, but the shock 
is avoided by the gradual action of the friction brakes. 

By using the author's device, shown in Fig. 717, at 3, the 
value of o might be maintained constant, or by proper construc- 
tion of the guides the wedge friction pawls, similar to Fig. 718, 
may be used ; the blocks acting on both sides of the guide. 
This would reduce the stress upon the frame very materially. 

The system of brakes used upon railway trains are really 



forms of friction checking ratchets. The shocks due to suddeE 
stoppage are also to be avoided, and if the wheels are braked 
too firmly the sliding action is simply transferred to the rails. 

\ 254- 

Continuous Running Ratchets. 

Continuous ratchets (^ 235, No. 4) consist of such combina- 
tions of pawl mechanism as act to drive a member in a given 
direction with practically a continuous determinate motion. 
This may be effected by combining two single running ratchets 
in such a manner that they both act upon the same wheel, one 
pawl attached to the arm c, which is stationary, the other swing- 
ing about the axis i, Fig. 749, this being a very common form. 

a, b. 







\ ^ 


■^^L ._n 7r 


Ol 


\ T r^ 


■ 




Fig. 749- 

In this case 3 . 2 is the checking pawl, and 3' . 2' the driving 
pawl. A movement of the driving pawl, if a little more than 
one tooth space, moves the wheel one tooth ; a little more 
movement than two spaces moves it two teeth ; and a regular 
back and forth motion gives a forward movement at intervals of 
a single pitch space. 

If this device is made with step ratchets, as in ? 243, the pitch, 
may be subdivided into 2, 3, 4 or more parts, and for some pur- 
poses, such as saw-mill feed motions, this is very desirable. 

If the arm which carries the feed pawl swings about an axis- 
4, removed from i, Fig. 749 b, there will be a movement be- 
tween the pawl and the poiut of application 2 on the wheel ; 
while in the arrangement shown at a the motion of the two is 
identical, and hence no wear occurs. 

The two pawls may be connected so that both of them be- 
come drivers. If they are arranged so that their movement is 
alternate, as in Fig. 750 a, the wheel will be moved forward for 

a. b. 





Fig. 750. 

the movement of the lever in each direction, giving a double- 
acting ratchet motion, the so-called Lagarousse Ratchet.* This 
may be also accomplished in various ways, as in Fig. 750 b. 
For any movement of the arm which is less than i and more 
than ;-2 the pitch, the wheel will be moved i pitch for each 
vibration, and hence for a half vibration a feed of a half tooth 
may be obtained. Step pawls may also be used with these de- 
vices to obtain further subdivisions. 

If, in Fig. 750 a, we hold the lever c-^ c^ rigidly, and instead 
permit the arm d to vibrate with the same angle about the axis 
4, the wheel moving with it, we obtain the same relative feed 
motion. t This has been used by Thomson in a telegraph, 
apparatus. 



* Berliner Verhandlungen, 187^, p. 345. Prize essay by Dr. F. Nitzsch, on 
Safety Checking Devices for Mining Apparatus. See aiso Mairs, Berg und 
Huttenmann, Z., 1879, p. 361. 



* Named from the inventor, M. de la Garousse, and used in 1737. Belidoi^. 
Arch. Hydraulique. 
t This' is the ordinary kinematic inversion. 



THE CONSTRUCTOR. 



165 



A continuous ratcbet gearing may be so arranged that back- 
"ward movement of the wheel is utilized to compel a uniform 
division of motion. 

This is the case with the feed motion used by Gebriider 
Mauser, of Oberndorf, in their revolvers, Fig. 751. In this case 




Fig. 751. 



Fig. 752. 



a crown wheel is used (see Figs. 677 and 67S). The wheel is at 
■a ; b'xs the feed pawl, jointed at 3 to the slide r, the whole being 
carried in the frame d. The zig-zag profile is formed in the rim 
of the crown wheel, one portion being parallel to the axis, the 
other spirally inclined, so that the angle of thrust is ct < 90° — 
<^ and > (? 237, cases 4 and 5). The movement of the pawl 
produces a backward movement of the wheel. It should be 
noted that at 1' and 2" steps are made in ends of the tooth pro- 
files in order to guide the pawl into the proper path and keep it 
from reversing. 

The anchor ratchet of Fig. 6S2 may be used for a feed motion, 
as in Fig. 752, in which there is also the reverse action of the 
wheel, in accordance with the notation of ? 237. Here the wheel 
is at a and the anchor at b' b" . When the latter is moved into 
the position shown by the dotted lines, the wheel is moved 
backward yi pitch, and the return vibration completes the pitch 
movement. In order that the anchor shall enter the teeth pro- 
perly, the movement should be quick, especially at the entrance 
of the pawl into the space. This is well obtained by electro- 
magnetic action. 




Fig. 753. 

?255. 
Continuous Ratchets with Locking Teeth. 
If it is desired to use ratchets according to the method given 
in Fig. 749, additional parts must be devised to move the pawl 





in and out of gear. A simple method of accomplishing this re- 
sult is to use a single tooth wheel for the driver, and operate 
the pawl in the same mauner as in Fig. 753. 

Before the single tooth 5 begins to drive the wheel a, the arm 
6 lifts the pawl b and lowers it into the next space just as the 
tooth ceases to drive. In this case the usual gear tooth profiles 
may be used. Still better is the "dead" tooth profile of Fig. 
754, in which the entrance and withdrawal of the pin tooth 
both lock the wheel while the pawl is being lowered. 

This form may also be used for rack feed movement. Fig. 755. 
In this case the profile of the pin tooth is formed in several 
arcs ; 2' 2'" being struck from 3, and 2" 2.'" and 2' 2'^ being 
the paths of the corners of the space (see \ 203). 

By using the cylinder ratchet, as shown in Fig. 696, the num- 
ber of parts can be reduced, since the driving gear and check- 
ing pawl may be combined in the same member. The resulting 
forms, Figs. 756 to 75S, are variousl}- called : Maltese Cross ; 




Fig. 756. 



Fig. 757- 



Fig. 758. 



Geneva Stop, used in Swiss watches, in which case one of the 
tooth sections is filled out ; or after Redtenbacher we may call 
them single tooth gears, although this is hardly correct, for the 
general form of Fig. 75S may have several teeth, and a second 
tooth is dotted in Fig. 756. 

A great number of variations may be made of these cylinder 
ratchet motions. An interesting form is the intermittent gear- 
ing of Brauer, Fig. 759.* 




Fig. 759- 



Fig. 760. 



The pinion a is the driver, and the wheel b is driven, and 
between the passage of each tooth of the pinion the driven 
gear remains stationar3' for a short space, about i of the pitch. 
The points of the teeth of the driven wheel here act as ratchet 
teeth, in a similar manner to the arc of repose of the single 
ratchet gearing of Fig. 756. 

The cylinder ratchet gearing of Fig. 760 is similar to that 
shown in Fig. 700, and is used in the counting mechanism of 
English gas meters. In Fig. 761 is a modified spiral ratchet of 





Fig. 761. 



Fig. 762. 



the same general type as Fig. 702, with only a portion of the 
path of (5 in a spiral, and a similar variation of Fig. 704 is shown 
in Fig. 762. 



Fig. 754- 



Fig. 755- 



* Royal Germau Patent, No. 5583, 1S7S. 



i66 



THE CONSTRUCTOR. 



1 256. 
Locking Ratchets; 

Locking ratchets include all the numerous devices by which 
the parts of a mechanism are firmly held against the action of 
external forces, and yet readily and definitely released when 
desired (see i 235, No- 5) ; thus the various clutch couplings are 
included, also car-couplers and similar devices. 

Locking ratchets occur frequently in the mechanism of fire- 
arms, especially to prevent the danger of premature discharge, 
etc. The great refiuements which have been introduced in such 
■weapons during the last ten years include especially the appli- 
cation of various forms of ratchets. The following single in- 
stance will serve to illustrate : 

The mechanism of the well-known Mauser revolver may be 
divided into two series; one to effect the discharge and the 
other to unload or remove the empty shell from the chamber. 
The first may be called the discharging mechanism, the second 
the unloading mechanism. We then have the following details : 

A. Discharging Mecha]iisin. 

This includes the revolving chamber, barrel, hammer, spring 
and accompanying smaller parts, giving as combinations : 

1. Hammer, spring-rod and trigger = ratchet rack, as Fig. 

659. 

2. Spring-rod and trigger, acting as locking ratchet for the 

above, as Fig. 664. 

3. Spring-rod, pawl and revolving chamber = continuous 

ratchet with crown wheel and bolt pawl, as Fig. 751. 

4. Securing pawl and revolving chamber = locking ratchet, 

as Fig. 677. 

5. Revolving chamber and pawl, forming a ratchet gearing 

with limited travel. 
5. Tumbling ratchet and securing pawl = ratchet gearing 
for three positions. Fig. 669. 

7. Catch on the axis of hammer ^ locking ratchet, as Fig. 

695. 

8. Trigger guard and pin = locking ratchet and stationary 

pawl. 

9. Checking-plug and trigger ^ locking ratchet with sta- 

tionary pawl. 

10. Rifled barrel and bullet = screw and nut. 

B. Unloading Mechanism. 

This includes au axial slide which catches under the rim of 
the empty cartridge shell to withdraw it, actuated by a toothed 
sector and revolving clamp and axis called the ring clamp. 
These include the following combinations : 

11. Unloading slide and sector ^ slide with rack and pin- 

ion. Fig. 381. ^ 

12. Axis of revolving chamber, with pawl to prevent end- 

long motion, ^ locking ratchet gear, as Fig 695. 

13. Ring clamp, barrel and chamber bearing = locking rat- 

chet gear with stationary pawl, as Fig. 654. 

14. Ring clamp axis and axis of securing pawl = locking 

ratchet, as Fig. 701, forming with (13) a locking ratchet 
gear of the second order. 

15. Ring clamp axis upon the reverse motion of the ring 

clamp forms, with the axis of the securing pawl, a 
locking ratchet .gear, which combines with (4) to form 
a similar gear of the second order. 

16. Securing pawl acts as a catch for the axis of the ring 

clamp in the axial direction to form a locking ratchet 
gear, as Fig. 695, forming also with (4) a similar gear of 
the second order. 

17. Ring clamp hub and axis of securing pawl = locking 

ratchet, as Fig. 695, and with (4) gives one of the 
second order. 

This analysis shows that in the Mauser revolver there are 17 
mechanical combinations ; these are composed of 26 pieces. 
Classified, these are as follows : i releasing ratchet, i continuous 
ratchet, 2 driving ratchets, 11, locking ratchets, of which four 
are of the second order, i screw motion and i slide motion. 

A very important application of locking ratchet mechanism 
is found in the signal apparatus of Saxby & Farmer for use on 
railways, and made in Germany by Henuing, Busing and others. 
This includes many ratchets of higher orders, reaching to the 
tenth, twelfth, or even higher. When this is used in combina- 
tion with the electric systems of Siemens & Halske, as in the 
block system, we have the further combination of two systems 
of the higher order with each other. 

A branch of locking ratchets which exhibits a great variety 
of applications is found in the different kinds of locks, such as 
are used for securing doors, gates, chests, etc. These extend 
from the most primitive forms, made of wood, to the most re- 



fined productions of exact mechanism, and their study possesses 
an historic and ethnographic interest in addition to their me- 
chanical value. 

A door forms itself a ratchet combination ; the door being 
the part b, the strike the part c, and the bolt or other piece 
which keeps it from being opened is the part a ; doors with 
latch bolts being running ratchets, and doors with dead bolts 
being stationary ratch- 
ets. A simple lift latch 
and door, as the furnace 
door shown in Fig. 763, 
is really a section of 
a crown ratchet wheel 
with running ratchet 
gearing. 

A door with sliding 
dead bolt, as used on 
common room doors, is 
a similar section of rat- 
chet gear with station- 
ary ratchet. ■ 

In key locks, the key 
is the releasing member 
of the ratchet train, and also serves to actviate the bolt after it 
is released. The key and ratchet mechanism are arranged in 
most ingenious manners, so that numerous permutations can be 
made to effect the release- 
Some of the most important systems of lock construction are 
given as examples : 

Example I. — The common so-called French lock. Fig. 764, is similar to the 
ratchet of Fig. 753. The bolt is a sliding rack, the " tumbler " b being often. 





Fig. 764- 



FiG- 765. 



as in this case, made in one piece with its spring. The case of the lock cor- 
responds to the frame for the ratchet mechanism, and the key acts as the 
releasing and actuating member. 

Example 2.— T)\& Chubb lock. Fig. 765, which is always made with a dead 
bolt, forms with the door and door frame a ratchet gearing similar to Fig. 
69E. Tne bolt is secured by means of several ratchets of precision, as in Fig, 
706, and is moved by a ratchet as Fig- 755- The key, the axis 4, and the vari- 
ous bittings of the key form a system of pawls. The whole is a ratchet sys- 
tem of the second order with precision gear. 

Example J. — The Bramah lock. Fig. 766^ and Fig. 7666, is differently con- 
structed- In this case the dead 

Jcg bolt is actu.ited through the 

medium of a cylindrical driving 
ratchet , gear, which does not 
contain the mechanism of se- 
curity, the latter being in a 
distinct portion of the lock. 
Fig. 766 A. This consists of a 
number of sliding precision 
pawls, as Fig- 707, the number 
being 6 to S (in the illustration 
5)- '1 he member a of Fig. 707 
is here made in the form of a 
ring with internal teeth, se- 
cured to the escutcheon a by 
screws- The key is a prismatic 
adjuster of the slides, and the 
whole is a locking mechanism 
of the third order with ratchets 

of precision. The spiral spring around the pin restores the slides to their 

extreme position when the key is withdrawn. 





lo 




I 


lo 


TT 




i 

"0 i 






-■0 


& 












^ 


5^ 


)A 



Fig. 766 a. 




Fig. 766 d. 

£xam//e 4.— The Yale lock, Fig. 767 a and d. is also a system in which the 
mechanism of security is separated from the bolt mechanism. This is 
again a system of the third order, with ratchets of precision. The key is a 
flat prism, (corrugated in recent locks) and serves to place precision bolts, or 



THE CONSTRUCTOR. 



167 



piu tumblers in proper line, and also operate the bolt. The figure shows 
the method of connecting the cam b^ to the plug a. 

The so called combination locks are locking ratchets with precision pawls, 
operated without a key; by being placed successively in the positions for 
release in accordance with a previously selected series of numbers and dial 
marks. 



h ; 1 








1 lo 




_^r 


'?V'-" 




^ 


-i — . . 






Fig. 767. 

The numereus systems of Arnheim, Ade, Wertheim, Kleinert, Polysius, 
Kromer, and others are mostly locking ratchet systems of the fourth order, 
or combinations tliereof The American manufacturers, especially the 
Yale and Towne Manufacturing Company of Stamford, Connecticut, have 
shown great ingenuity in this industry.* 

1 2:7. 

Escapements — Their Varieties. 

Escapements may fairly be considered as among the most im- 
portant mechanical devices, since it is by their means that the 
elementary forces are used to regulate mechauical work. For 
this purpose they are used in the greatest variety, all forming 
ratchet devices in which the driven member is alternately re- 
leased and checked. The arc, angle or path through which the 
driven member passes between the interval of release and check 
is called the "range" of the escapement. During the passage 
over this range there elapses a definite amount of time, which 
maybe called the "period " of movement of the escapement. 
This is followed by an amount of time when the driven member 
is stationary, called the period of rest. The sum of the two 
forms the "time of oscillation." The range and the period of 
oscillation may be [a) constant, {b) periodically variable, or (..-) 
variable at will. 

We therefore have 



Uniform escapements. 
Periodical " 
Variable " 



and these will be briefly considered. 

I 25S. 
Uniform Escapements. 
If, in ordinary running ratchet, Fig. 76S, we have the wheel a, 




impelled by a weight or other force, and suppose the pawl b, 
lifted and dropped quickly, as by the arm b-i, the wheel will 
move one space, and an escapement will have occurred. In 
this case the range will be one pitch. If, after a definite time, 
this operation is again and again repeated, we shall have a 
uniform escapement. In mechanism the releasing and check- 
ing action is produced mechanically and not by hand, the im- 
pulse being obtained from the movement of the wheel. 



* The ancient and modern Ej^yptian locks, also those of ancient Greece, 
Rome, India and China, contain the principle of running ratchets with flat 
pawls, actuated by a key pushed directly into the lock. The Egyptian lock, 
with pin precision pawls, is quite similar to the Yale lock in principle, al- 
though very ditferent in construction. Ancient Roman locks, found in 
Pompeii, are similar in principle. Wooden locks are still in use in China, 
Persia, Bulgaria, Russia and Southern Italy, also in the Faroe Islands and 
Iceland. At the suggestion of the author. Professor Wagner, of Tokio, suc- 
ceeded in inducing some Japanese lockmakers to make a very complete and 
intelli^bie collection of native locks for the kinematic cabinet of the Royal 
Technical High School at Berlin. 



The most general examples of uniform escapement are found 
in watches. In these impulses are isochronous, and obtained 
from the inertia of a vibrating body. The wheel a is called the 
escape wheel. The vibrating member, or balance wheel, makes 
its oscillations in nearly equal times for great or small vibra- 
tions. If, therefore, in a watch escapement, the time of the fall 
of the pawl is less than the time of oscillation, the most impor- 
tant requirement is fulfilled, namely, that for uniform periods 
of time the same number of teeth of the escape wheel shall pass, 
and the corresponding angle may then be used as a measure of 
time. A given amount of work may ajso be abstracted from the ^ 
motive power and used to produce the impulse. These impor- 
tant points have been fulfilled in the design of escapements, 
and it has been made possible to measure time with a great 
degree of accuracy. When the highest accuracy is demanded 
the greatest care must be given to the construction and execu- 
tion, and to the reduction of friction and compensation of the 
balance. 

In the case of watches the duty of the impelling force is 
simply that of overcoming the resistance of the mechanism, 
the function of the escapement being to provide against any 
acceleration of the rate motion, and the impulse which is re- 
quired to operate the escapement may be considered as a por- 
tion of the resistance of the mechanism. 

A systematic discrimination between the various kinds of 
watch escapements will show that they vary as to the checking 
ratchet device, the impelling device, the release and the accel- 
erating device. We may have Simple or Compound escape- 
ments of the lower or higher orders. Some examples are here 
given. 

A. Simple Escapeiiienti. 




Fig. 769. 

Example /.—The Free Chronometer Escapement (Jullien le Roy, Earn- 
shaw, Arnold, Jiirgensen), Fig. 769. The running ratchet gearing a, b, c, is 
similar to Fig. 76S. The pawl b is provided with a flat spring 3. The im- 
pelling device is the balance wheel rf, which acts as a pendulum. The re- 
leasing device is at 4 . 5, and is attached to d, and when it swings to the left, 
impelled by the movement of the watch, it releases the pawl by means of a 
second running ratchet at 5. .^t c' is a stop for the pawl b. At 5' is the ac- 
celerator which, for each tooth of the escape wheel «, swings from 5' to 5". 
As it returns, the pawl I* engages with the tooth which has just left the point 
5". The spring b' permits the releasing tooth 5 to pass back dnring the 
return oscillation. Tlie balance wheel can swing freely beyond 5" and back 
without engaging with the escape wheel, hence the name " free " escape- 
ment. ■'■ 

Example 2,— The Duplex escapement, Fig. 770, is derived from the ratchet 
of Fig. 699. The escape wheel is upon the same axis as the checking pawl 





Fig. 770. 



Fig. 771. 



* This beautiful movement is apparently the first form which was applied 
as a pendulum escapement, having been used by Galileo in 1641. 



i68 



THE CONSTRUCTOR. 



I> \ the accelerator is at 4, acting upon the impelling pawl at every vibration 
"between 4 . 4'. 

The so-called "verge " escapement is similar in construction, except that 
the arm b' is longer and curved. The simplicity of this form as compared 
Tvitli the preceding is due to the fact that the impelling and checking pawls 
are made in one member It will be noticed that the entrance of the tooth 
■of the escape wheel into the space, causes a slight reverse movement at a, 
due to the fact that b is really a tumbling ratchet gear. This escapement has 
been called duplex: by its Knglish inventor, although some contend that it is 
properlv a double wheel escapement, although the two wheels are combined 
in one. 

Example 3.— KwoW^^x method by which the checking and impelling pawls 
may be combined is shown in the Hipp escapement, Fig. 771. This consists 
■of a simple running ratchet a, /•, c. The pawl 6 is a plate spring, which is 
lifted and dropped by the passage of the teeth. The acceleration is given by 
the deflection of the spring. If the impelling force upon the wheel a is 
great, two teeth will pass, but this can be detected by the note emitted by 
the spring, which will then be one octave higher than before. 

B. Compound Escapeinenis. 

Example 4. — Lamb's escapement. Those escapements which have two 
escape wheels are properly classed as compound, and to this class belongs 
I^amb's escapement. This consists of a running ratchet gear, similar to 
Kxauiple I, and the same form of impelling device, but between these is an 




internal wheel with pitch ratchet gearing, similar to Fig. 6S6, which is im- 
pelled with each direction of vibration. Another double-wheel escapement 
15 Enderlein's, based on Fig. 70-, also one devised by the author, like Fig. 6S6. 

Example 5.— ls\\x^%^' & Escapement (also invented by Tiede), Fig. 772. This 
is a double ratchet gear system, with one pawl in compression and one in. 





Fig. 773. 



Fig. 774. 



are moved alternately by the pendulum ;' for example, the arm b\, being 
moved into the dotted position, lifts the pawl out of gear, and the weight of 
the pawl and arm (sometimes assisted by a spring), gives an impulse to the 
return vibration of the pendulum, the acceleration being provided by the 
escape wheel acting on the portion //'. A similar action takes place on the 
other side. 

Example 6. — Bloxam's or Dennison's so called "gravity'' escapement, Fig. 
773. The escapement is controlled by a pendulum suspended by a spring at 
4. The escape wheel is made in two parts, as Fig. 686. The accelerating 
surfaces //' and //" are much better arranged than in the preceding exam- 
ple, the friction being reduced. A fan is used also, as shown at e, for the 
purpose of preventing great acceleration of the escape wheel, which might 
otherwise occur in the large angle (60°) of escape. The fan is not fast to the 
axis of the escape wheel, but connected by running ratchet so that its mo* 
mentum. is not checked as the escape wheel is stopped. 

Example y.—Fvee Anchor Escapement, Fig. 774. The two pawls are com- 
bined into one anchor, as in Fig. 682, and the action is much the same as 
Fig. 772. The escape is controlled by a balance wheel at tf. The pawls 2' 
and 2" are operated through the arm i% and at the same time the impulses 
are given by the action of the escape wheel upon the inclined surfaces//' 




and //". The pawls are technically known as pallets. The tooth action at 
5 is a continuous ratchet gear similar to Fig. 754. The arm 1^3 is limited in 
travel by pins at 3' and 3", or in some forms by a fork at 4 Since there is a 
ratchet at 5 and also at 2, this forms a system of the second order.* 



(^^=f^ 




Fig. 776. 



tension, ^1 and i^- At 2' and 2" is a "dead'' pawl action for checking, and 
.at //' and //'' a running pawl action for impelling. (See Cases 5 and 7, § 237). 
The pawls are lifted by the pendulum d. The releasing arms 3' . 5' and 3". 5" 



* A watch escapement of the third order has recently been designed by A. 
E. Miiller, of Passau. This is made with a cylindei ratchet, as Fig. 699^, 
between the arm and the escape wheel. 



THE CONSTRUCTOR. 



169 



Example 5.— Graham's Escapement, Fig. 775. The construction is very 
similar to the preceding. The connection 5 between the anchor-arm A) and 
pendulum d, is different, and the arm b-^ does not come to rest, but both it 
and the pallets 2' and 2" slide upon the teeth while the escape wheel is 
stopped. An earlier form of pallets for this escapement is shown at b'\ and 
b\ (called Clement's Anchor, frona Clement, 16S0 ; but described by Dr. Hooke 
in 1666). This form produces a brief reverse movement to the escape wheel 
at each oscillatiou. 

Example g.—Th^ form of ratchet of Fig. 6S4 is used in Lepaute's escape- 
ment, which was really invented b}- the watchmaker Caron, afterwards 
Marquis Beaumarchais. 

Example jo. — Cylinder Escapement, Fig. 776. This is made from the 
cylinder ratchet of Fig. 700, the impelling surfaces being divided between 
the anchor and the teeth of the escape wheel. The cylinder b is attached to 
the axis of the balance wheel, and the wide spacing of the teetli of the escape 
wheel permits a correspondingh' wide amplitude of oscillation. If we im- 
agine the pallets of Graham's anchor to be formed between two concentric 
circles (as, indeed, most watch Tiakers construct them), the " cylinder'' will 
be seen to be a similar anchor. 

Example II. — Crown Wheel Escapement, Fig. 777. Escapements con 
structed with crown ratchet wheels (5 z\\) are the oldest forms used iu 





Fig. 777- 



ratchets.* The form of the pallets causes a reverse movement, and in the 
old watches using a balance with its centre of gravity in the axis of oscilla- 
tion, without any assisting spring action, this reverse movement was a 
necessity, which accounts for the long and extended use of this form of 
escapement. Toward the end of the fifteenth centur}' the hair spring was 
introduced by Hele. in the form of a hog's bristle, and in 1665 Hayghens 
made the steel hair spring, which made the construction of the modern 
chronometer possible. The crown escapement is easily modified so as to 
remove the reverse action, as was done by the author in 1864. We then have 
a "dead" tooth action, as Fig. 699. The modified escapement is shown in 
Fig. 778 ;, the pawls are practically hyperboloidal in form f 

C. Poiuer Escapements. 

Ill the case of watch escapements the impelling force is only 
used to overcome the resistance of the watch mechanism. 
Escapements can also be used to regulate greater forces, such 
as are intended to perform useful work, and these may be 




Example /r.— Power Escapement for a Reciprocating Movement, Fig. 779. 
At a by Cx and a b.> Co are ordinary running ratchets, the pawls b\ and b^ of 
which can be released and engaged by suitable auxiliary mechanism. This 
mechanism is either a substitute for or identical with the legulating device 
(balance wheel, pendulun, etc ) of a watch escapement. The escapement is 
intended to control the motion of the swinging arm Cby means ot the lever 
^i and tlie descending arm -^i- This is accomplished by a double acting 
ratchet system di dq 5 (as Fig. 671), by means of the slide e, "driven from 8 by 
the arm cj. 

The action is as follows: When the parts are in the position shown in the 
figure, the motion of the wheel a to the right moves the arm ci by means of 
the pawl bi until the trigger 10" trips the pawl d.^ and shifts the engagement 
at 5 into the position 5' (in the small figure to the left). This action, by 
means of the trigg-er at 6", throws in the pawl Aj and stops the wheel a. At 
the same time bi is thrown out of gear by the connections d^, 6' and 7', and 
the counterweight Q returns the arm ci to its original position. This brings 
the trigger 10' against the lever di, and again shifts the engagement at 5. 
The pawl b] falls into gear, and the pawl b.^ is disengaged, leaving the wheel 
a free for another forward movement. 



The preceding escapement can be readily converted into a 
double acting one by introducing a second ratchet wheel toothed 
iu the opposite direction, with proper pawl on c^ and trigger 
connections to d^ ; the other portions would remain the same. 
This escapement appears to be new, and many important appli- 



§ 258. 

Periodicai, Escapements. 

A great variety of periodical escapements are to be found in 
the striking mechanism of clocks and repeating watches. The 
entire period is the revolution of the hour hand, and if the half 
hours are struck the order will be 



4, 



making iu all 90 strokes iu the twelve hours. A fan regulator 
is used to cause the strokes to follow each other uniformly. 
There are two systems of escapement in use for this purpose, 
the German and the English, the latter also used for repeaters. 
An essential piece of the latter, the so-called "snail," has been 
shown in Fig. 688 ; its function is to control the number of 
strokes. Further subdivisions cannot be here discussed, but it 
must be remembered that the striking arm is itself a ratchet 
mechanism.* 

Important applications of periodical escapements are found 
in the self-acting spinning mule, and both these and the clock 
striking mechanism are examples of power escapements. 

The mechanism in Piatt's mule is here briefly shown. Fig. 
780, a aud b. The shaft l is required to make rapid turns 




Fig. 779. 

called power escapements. Alarm and striking clocks are of 
this class, and there are numerous other forms. The following 
example will serve to illustrate : 



* This has been used since the tenth century, having been invented by 
Bishop Gerbert, afterwards Pope Sylvester II, about 990 ; also by Hcinrich 
von Wyck about 1370, and applied to a pendulum hv Huyghens. The oldest 
tower clock in Nuremberg, built about 1400, has such an escapement- 

t In the Kinematic cabinet of the Royal Technical High School there is a 
schematic series of models of clock and watch escapements. 



through 90° at intervals of different lengths of time. The wheel 
a is an escape %vheel with teeth in four concentric riugs, I, II, 
in, IV (compare Fig. 686), each ring having one tooth. The 
other side of the wheel a is shown in Fig. d, where is the rat- 
chet chain a d e. When a is released, the pressure of d at 5' 
moves it slightly and brings the running friction wheel e into 
contact, thus driving a through a quarter revolution, toward the 
close of which the pawl (/ again enters iuto engagement. 



► See Ruhlmann RMtenbacher. Denison. 



17 o 



THE CONSTRUCTOR. 



The recesses in a permit the friction wheel to run free when 
a is at rest. This is evidently a form of ratchet gearing in it- 
self. The order of escapements at 2 is as follows : 
I II, II III, III IV, IV I. 
This is controlled by a second escapement, shown in Fig. 781. 



We have already intimated that the various forms of coup- 
lings may be considered as varieties of ratchet gearing- The 




Fig. 7S1. 

The pawl b of Fig. 7S0 is connected by the rod/ to the beam a, 
as shown. This mechanism is a step ratchet of four steps. The 
steps are the pawls ip b.^, b,, and the stop on the frame c ; giving 
the positions 21, 211, 2m, 2iv. The action takes place in the four 
following periods : 

1. Drawing and spinning — a checked at 21 

2. Stretching and twisting " " 211 

3. Holding aud spun thread " " 2111 

4. Winding and returning " " 2iv 

The succession of movements is as follows : At the termina- 
tion of the first period a projection on the carriage strikes the 
pawl b-i at 5'. The step lever, which is heavier on the right end 
than on the left, moves from position I to position II, in which 
it is held by the pawl b.,\ this, bj' means of the rod /, places the 
pawl b of Fig. 780 in the position 3 II, thus starting the second 
period. 

At the close of the second period the pawl b.^ is released, the 
lever falls to the position III, shifting the pawl 5 to 3 III, and is 
held by the pawl 63 at 2'". 

The third period, which is very brief, is terminated by the 
winder striking 5'", releasing the pawl b-^, and the lever as- 
sumes the position IV, and the rod f moves the pawl b into the 
position 3 IV, and the fourth period begins. 

During this period the carriage returns, and just before the 
close of its motion a roller acts upon the portion 5°, bringing 
the lever back into the first position. This returns the pawl b 
to its original position 3 I, and the succession is repeated. 

The entire mechanism forms a periodical escapement of the. 
second order, or, when the connections are included, the third 
order, and when taken together with the ratchet gearing, of the 
fifth order ; while a sixth ratchet mechanism is used for the 
primary control. 

I 259. 

Adjustabi^e Escapements. 

An escapement can be so arranged that the checked member, 
after the release, will again be checked by the impulse of its 
fresh start, thus forming what may be called a self-acting 
escapement. In a mechanism of this kind, the amplitude of 
the escapement is dependent upon the amount of displacement 
which is permitted to the releasing member. This may be 
made greater or less, and hence such devices maj- be called 
adjustable escapements. These devices are likely to play an 
important part in modern machine design. 

A simple form of adjustable escapement is shown in Fig. 782. 
This apparatus, designed by the author, is based upon that of 
Fig. 674. The ratchet wheel a is stationary, being fastened to 
the frame a' ; the pav*! is at b, and the link is in the form of a 
disc f, driven by a force C, and checked by the escapement. At 
3 . 5 is the guide for the pawl. This can be adjusted b}' the 
wheel d, by turning the latter more or less in the direction in 
which c is impelled. If d is turned so far that the pawl b is 
lifted out of gear, the force at Cwill set the disc c in motion. 
This latter carries with it the axis 3 of the pawl, which, by the 
action of the guide 5, draws the pawl into engagement again, 
entering the space 2 and checking the disc. In order to avoid 
an uncertain or irregular action, a brake may be used as at a". 
If the wheel d be moved forward regularly through two, three, 
or four arcs, the disc iTwill be released and checked successively 
in similar manner. 

'It will be evident from the foregoing that the ratchet gearings 
which form the foundation of the various kinds of adjustable 
escapements are so varied that the different constructions which 
ma)' be vised are very numerous. Among them may be men- 
tioned those in which friction ratchets are used, these posses- 
sing the advantage that the arc of motion of the escapement 
may be varied from the smallest to the greatest without being 
dependent upon any especial pitch. 




Fig. 782. 

same is true of the present subject. If it is desired to use this 
adjustable escapement as a disconnecting coupling, the follow- 
ing arrangement may be adopted : 

The disc c can be attached to the shaft which is to be set in 
motion, and the wheel a to the driving shaft, which is supposed 
to be in continuous revolution and is to be coupled to c. The 
teeth are then to be so arranged that by the revolution of a the 
pawl b, disc c and wheel d will be carried around together. 
When the disconnection is to be made, it is only necessary to 
hold the wheel d from revolving. The pawl-axis 3 will then 
move on and cause disengagement of the pawl at 2, and the 
disc c will come to rest. If the wheel d is then turned a short 
distance in the direction of rotation the pawd will .again, be 
thrown into gear and the parts once more connected. A coup- 
ling thus formed from an adjustable escapement may be called 
an adjustable coupling. 

The suitability of the application of toothed ratchet gearing 
for this purpose is open to question, and indeed toothed gearing, 
is only to be recommended for the lightest service of this kind. 
In most cases, if indeed not all, friction couplings are much 
better. An adjustable friction coupling is to be seen b)' refer- 
ence to Fig. 44S, in which A is the friction wheel, B is the pawl, 
disguised in the form of a cone, and b is the adjusting member. 

If a combination is made of an adjustable friction coupling 
with some form of transmission to a machine, such as a rope 
or belt gearing, so that it is thrown into action when any re- 
verse motion is attempted, we have what may be termed an 
automatic friction brake.* 




Fig. 783. 



*See German Patent, E. Langen, No. 21,922. 



THE CONSTRUCTOR. 



171 



Example. — Figr. 783 shows such an automatic, brake device as applied to 
the pontoon bridge at Cologne. At a is a friction cone combined with a spur 
gear a', driven by the shaft and pinion rt" in the direction to wind up the 
cord on the drum c'. The drum is fast to the chaft c, but the cone a is loose 
on the shaft. The wheel a is connected firmly to the shaft t:, when the cone 
b, which slides on a leather, is forced into engagement witii it, and this en- 
gagement is effected by the differential screw d and hand wheel d' . The use 
of the differential screw enables the equisite pressure to be obtained, and 
also causes tiie motion of d' to be in the same direction as c' when lifting. 
The friction of tlie cones binds the parts firmly together, so that a is practi- 
callj' secured to the shaft until d' is revolved backwards, w^hen c' follows by 
the action of the weight C, the cones slipping upon eacli other and the 
pressure being automatically regulated, and the motion at once checked 
when d' is stopped. 

Other and most important applications of adjustable escape- 
ments will be given hereafter. It maj', however, be here noted 
that by means of such mechanism the most powerful combina- 
tions may be controlled with the exercise of a minimum effort. 

\ 260. 
Generai, Remarks upon Ratchet Mechanism. 

Ratchet mechanism, as already discussed, is applicable to a 
most extensive range of uses ; in this respect far excelling every 
other form of mechanism. This is plainly due to the fact that 
ratchets are suited either to produce the effect of relative motion 
and relative rest. Considered in this light the six preceding 
classes may be grouped as follows : Common ratchets, checking 
ratchets, and locking ratchets are those which act to hinder 
motion, while releasing and continuous ratchets, as well as 
escapements, act to produce definite motion. The motion pro- 
duced by ratchets is intermittent while that produced by the 
forms of mechanism previously considered, such as cranks, 
friction, or toothed gearing, etc., is continuous. Mechanism for 
continuous motion may be called "running gearing,"* and 
practically merges into ratchet gearing. The general province 
of ratchet gearing has only been partially covered in the pre- 
ceding pages, where such forms as may strictly be considered 
machine elements have been included. An exception might be 
made as to the allied forms of springs, some of which, indeed, 
were referred to. There is, however, a large number of machine 
elements of a different kind, which usually involve the continu- 
ous action of the operative forces in one direction ; these in- 
clude tension organs, such as ropes, belts, chains, etc., compres- 
sion organs, fluid connections, and many others, all of which 
are considered in the following chapters. It will be seen that 
these may all be so arranged as to be fairly considered ratchet 
devices also ; as belts or chains may become friction or toothed 
ratchet gears, and even the valves of fluid connections are really 
pawls. t 

The pawl mechanism must also be extended to include these 
classes of machine elements, and their limits thus greatl}' 
widened, especially in the case of pressure organs. Examples 
of this will be found in the pistons and valves of pumps, both 
for liquids and gases, which may act as checking or locking 
ratchets, or in hj'draulic motors and steam engines as escape- 
ments, and in gas engines, as escapements and continuous 
ratchets combined. Similar comparisons may be made of the 
ratchet principle in the use of accumulators for hj'draulic cranes, 
presses, riveting machines, and the like, and in the cataract for 
single acting steam engines we find a complete analogy to the 
ratchet. In these cases we have ratchet systems of the higher 
orders. The history of the development of these machines is 
really that of their pawl membeis. 

A very interesting example is that of Fig. 779, in which, if we 
substitute a flow of steam for the ratchet wheel, we have the 
arrangement of the single acting high pressure steam engine 
with Farey 's valve gear. The numerous modifications of escape- 
ment gear, which are included in the steam engine, have occu- 
pied the activity of designers down to the present time. A 
number of the more recent valve gears have been shown in \. 252, 
and similar devices are used on engines for steam steering gear, 
called by the French " moteurs asservis," and such gear also 
plays an important part in them;ihanism of some of the so- 
called ''fish" torpedoes. 

In this manner the applications of pawl ratchets may be ex- 
tended before our eyes and yet the limitations are not reached, 
and the further researches are carried the broader and more 
general does the scope of this division of mechanism become. 
Not only does it include fluid pressure organs, both liquid and 
gaseous in a strictly mechanical sense, as in the case of pumps, 
etc., but also when these are considered in a physical sense with 
regard to their internal stresses. This gives a branch which 
may be called "ph^'sical" ratchet trains, of which the steam 
boiler is the most important example. In this, when taken in 
connection with a pipe full of steam, and suitable valves for 
opening and closing, forming what has been termed a steam 



column,* we have undoubtedly a physical ratchet train in which 
the particles of vapor are considered as a physical aggregate, 
which from the higher temperature, are under higher stress. 
Another example of a physical ratchet train is the apparatus for 
operation by liquid carbonic acid which has been recently used. 

Electrical accumulators are also instances of physical ratchet 
trains, as well as some applications of galvanic batteries, the 
action taking place by make and break of electrical contact. 
The dynamo-electric machine also becomes a physical running 
ratchet and the electric motor a physicabescapement, the whole 
forming a phj'sical running gear train. 

Again we may consider a "chemical" ratchet train, such as 
coal or any fuel, which, during combustion, releases the energy 
which is stored in it. This may be utilized in numerous ways, 
but for our present considerations, mainly iti the production of 
motion. Chemical action is also included in hot-air engines, 
and in the operation of telegraph apparatus in a similar sense. 

We maj' consider the principal factors in a steam motor plant 
as portions of a ratchet chain, somewhat as follows : 

Chemical ratchet ^ combustion of fuel, 

Physical " = steam generator, etc., 

Mechanical escapement = steam cylinder and attachments. 
Mechanical running gear = crank shaft and wheel, 

these four uniting to convert the released energy into mechani- 
cal motion. If we consider a locomotive engine, we have added 
to this another running gear in the shape of the driving wheels 
and rails, while the train and wheels and journal bearings unite 
to form a combination of the sixth order. 

Another chemical train may be formed by the use of explo- 
sives, which are released either mechanically, as by percussion 
or friction, or chemically, by combustion of some auxiliary 
material. Again, we may have releasing gear of the first, second, 
or higher orders. 

In the case of most firearms the release is of the second onler, 
since the mechanism of the lock acts upon a fulminate by per- 
cussion, and the heat of the latter releases the pDwder. 

If we examine and classify all mechanism of transmission in 
the above manner, it will be apparent that all forms are included 
in one or the other of the following classes, viz.: mechanical, 
physical, or chemical ; these also entering into combinations of 
the higher orders with each other. 

The steam engine itself, as we have already seen, consists of a 
driving train of the fourth order. Trains of still higher orders 
are of frequent occurrence. 

In the recording telegraph, with relay, we have a ph5'sical 
ratchet train of the second order, releasing a mechanical run- 
ning train and operating a recording train, both physical trains 
actuated by chemical trains, the whole forming a combination 
of the fifth order. The ordinary si.gnal mechanism of a railway 
station, when mechanically operated, is a system of the fourth 
order. 

The Westinghouse air brake, not considering the boiler, is a 
train of the fifth order, consisting of an escapement (steam 
C5'linder), driving ratchet (air cylinder), intermittent ratchet (air 
vessel), escapement (piston and valve connections), friction 
checking ratchet (brake gear). If we include furnace and 
boiler, this becomes a train of the seventh order, and may be 
still further extended. 

A still more noteworthy example is found in the application 
of compressed air for the purpose of operating pumping ma- 
chinery at the bottom of deep mine shafts. In this case we 
have : 

1. Furnace = chemical ratchet train. 

2. Boiler • = physical " " 

3. Steam engine ^ mechanical escapement train. 

4. Shafting and transmission to " running 

5. Air compressor, " driving ratchet. 

6. Air chamber, " intermittent "^ 

7. Air cylinder in mine, " escapement train. 
fS. Water cylinder in mine, " driv'g ratchet " 

The preceding discussion and ilhistrations of the relationship 
existing between mechanical, physical and chemical trains shows 
the necessity of combining mechanical and technical research, 
and a complete mechanical training therefore in eludes these three 
branches, and also 'the later science of electro-mechanics. 
Modern methods of invention require research into all of these 
lines of science, and the constantl}' widening field of mechani- 
cal engineering is thvis extending its work, while at the same 
time gathering into systematic form the many branches of 
applied mechanical science. 



* See the author's Theoretical Kinematics, p. 
tion was originall.y made. 
tSee Theoretical Kinematics, p. 458 et seq. 



vhich this classifica- 



* See Theoretical Kinematics, p. 4Q3 

t The system of clocks operated by pneumatic pressure from a central 
station, designed by Mayrhofer, at Vienna, forms a combination of 33 dis- 
tinct systems. 



172 



THE CONSTRUCTOR. 



CHAPTER XIX. 

TENSION ORGANS CONSIDERED AS MACHINE ELEMENTS. 
I 261 

Various Kinds of Tension Organs. 

The various forms of ciacbine elements which have alreadj' 
been discussed, have been those which offered resistance to 
forces acting in any given direction, forming more or less rigid 
constructions. We now have a series of elements which are 
only adapted to resist tension, and which are very yielding 
under the action of bending, twisting or thrusting forces. These 
include a great variety of rope, belt wire, chain laelt and similar 
transmission devices, all of which may be included under the 
general term of Tension Organs. Their usefulness is limited 
.by reason of the fact that they have only the single method of 
resisting force, but at the same time the element of flexibility 
permits the use of one and the same organ to transmit power in 
changing directions, and hence gives rise to many useful com- 
binations. An especially valuable feature of tension organs in 
practice lies in the fact that many materials are excellently 
adapted for such use, and can be more economically applied. 

Fig. 262. 
Methods of Application. 

A distinction is to be made between " standing and running" 
tension organs. The first are those used to suspend weights' 
support bridges, also in the construction of many machine de 
tails. E.xaniples of such use are found in suspension bridges' 
pontoon bridges, hawsers, guj' ropes, standing tackle, etc 
Running tension organs are used in machine design in connec- 
tion with other machine elements principally for the transmis- 
sion of motion. 

Running tension organs may again be divided into three 
classes according to their action iu connection with other 
machine elements. 

According as they are used : 

1. For guiding. 

2. For winding (hoisting or lowering). 

3. For driving, this also being possible by winding and un- 

winding. 
Combinations of these applications may be made, either with 
or %vithout the use of standing tension organs. In order to 
understand the various applications it is desirable to consider 
some of the most important combinations, henre these will be 
briefly examined, 

abed e 



arc of contact. This action, which here opposes the motion of 
the cord, is in other instances made of great utility. Cord- 




^H^ 




?f 


^/^ 


:^^ 


{ 


i^ 


(0L l5) 


f /«\ IT 


li 


;: 


y \s 





Fig. 784. 

I. Guiding. — Fig. 7S4 shows several combinations, adapted 
solely for guiding. At a is the so-called stationarj' pulley, in 
which a cord, led off at any angle, is rised to raise and lower a 
load 0. The dotted lines show the position of guides, or in the 
absence of these the direction of motion is governed by the 
action of gravity. At b we have the so-called movable pulley, 
the pulley being combined with the moving piece ; the weight Q 
is here supported on two parts of rope. Form c is a combina- 
tion of a and h, and is the well known tackle block. Form d 
consists of four sets of form a, and the action of the cords com- 
pels the piece Q to maintain a parallel motion. This is practi- 
cally applied in Bergner's drawing board. 

In like manner four pulleys of form b may be combined as in 
form e. This is the old parallel motion for spinning mules, also 
used as a squarin.g device for traveling cranes.* 

The use of pulleys and bearings is to reduce friction at the 
point of bending, and roller bearings, as Fig. 566, are also used, 
but when the bending surface is well rounded the pulleys may 
be dispensed with. Fig. 7S5, at a, b, c, shows such arrangements, 
the action being the same as before, but with greater friction. 
The arrangement at rf is a six-fold cord, aod in sail making eye- 
lets are often used in similar manner, as at e. The friction is 
great in all such devices, because the cord presses hard upon 
the point of curvature ; its magnitude increases rapidly with the 




I1 




Fig. 7S5. 

friction, which is to be considered as a particular case of sliding 
friction, plays a very important part in constructions, involving 
tension organs, and will be more fully considered hereafter. 




In Fig. 786 is shown Riggenbach's rope haulage system for 
use on inclined trackways, or so-called "ramps." In this 
arrangement, the descending car is loaded at the top of the 
ramp with sufficient water to enable it to draw up the ascending 
car by the power of its descent. The speed can be controlled 
by the descending weight, and also a weight acting upon wheels 
gearing into a rack ir.f 

2. ll'iiiding. — The most important forms of winding gear are 




Fig. 787. 

shown in Fig. 7S7. At a is the common windlass, also known 
as a winding barrel or drum, extensively used in many forms of 
hoisting machinery ; i is a drum for spiral winding of a flat belt, 
the belt being wound upon itself, and side discs being provided 
as guides for the belt ; c is a spirally grooved drum for winding 
chain ; (/ is a conical drum, with spiral groove, used in clocks 
(there called a fusee), also for hoisting machinery with heavy 
rope ; and <? is a rope "snail" used on the self-acting mule, to 
produce the varied speed of the carriage. Many combinations 
of winding and guiding devices are made, also of winding de- 
vices with each other. 





Fig. 788. 

In Fig. 788 are shown several lowering devices. At a is a 
lowering drum for warehouse use ; the unwinding coil at ll\ 
lowers the load O, while the cord of the upward moving coun- 
terweight O., is wound on the drum at U'., ; a brake can be ap- 
plied at B, and when necessary, guide pulleys used as at L L. 
Form 5 is a lowering apparatus for coal trucks, consisting of a 
combination of two winding coils, with a brake at B. The 



* Form i/is a kinematic iuversioii of the older form e. 



f Numerous illustrations are in use in Switzerland and elsewhere, with in- 
clines varying from 25 to 57 per cent. 



THE CONSTRUCTOR. 



^73, 



couuterweight Q.^ is in the form of Poucelet's chain, the action 
being to vary the rate of descent of the load II'.,. This appa- 
ratus, which is called a "Drop," is much used in the coal 
mining districts in England. Form c is Althan's furnace hoist, 
and consists of two drums with steel bands. The load of water 
at (2i, by its descent, raises the charge C?2 to the top of the fur- 
nace, after which the water is drawn off, and the emptj' car de- 
scends and the water vessel is raised to the top again. The 
speed is controlled by a brake at B. 





Fig. 7S9. 

Wrapping connections bave been used from early times in 
connection with beams and levers, as shown in Fig. jSga, and 
the form 6 is especially applicable to scroll-sawing machines. 
Form cis a combination made with very fine steel bands, and 
used in the Emery weighing machine. 

Combination windlasses are frequently' used for lifting weights, 
some forms being shown iu Fig. 790, and other combinations 
also in complete machines for hoisting, as in Fig. 791. 



1 



w. 




W. 




=- 


'■=fftm 




-—- 




-U4iJ 












L 






Q 







Fig. 790. 

In Fig. 790, a is the so-called Chinese, or Differential Wind- 
lass, consisting of two windlasses and one sustaining combina- 
tion ; b is another differential combination used in a traveling 
crane designed by Brown, of Winterthur, the arrangement 
being intended to obviate the lateral motion of the load. 
Another arrangement for the same purpose is shown at c (de- 
vised by the author in 1S62) ; it consists of two drums united iu 
one. The signal arms and automatic safety gates, now so much 
used on railway's, are operated by a combination of winding and 
guiding members, chains being used on the winding barrels and 
wire connections on the straight lines. 

Winding and guiding members are much used in cranes and 
hoisting machinery, several combinations being given in Fig. 




791. A crane with boom of variable radius is shown at « ; b vs, 
a pair of shears operated by three windlasses, ]]\ and W„ for 
moving and holding the shear legs, /?<, for hoisting and lower- 




FlG. 792. 

ing the load ; c is a form of bridge crane, using a trolley in 
combination with two winches. If both winches are operated 



in parallel direction and uniform speed, trolley travel is effected, 
hoisting or lowering by unequal wind motion. 

In Fig. 792 a, three drums and one guide sheave are used ; b is 
made with four drums and two guide sheaves, a combination 
used in steering machinery for operating the tiller ; and c con- 
sists of two drums and two guide sheaves so arranged that one 
load is raised as tLe other is lowered, this being used in mine 
hoists. This is also used for inclines or "ramps." When the load 
is aUvaj's to be lowered, the descending load does away with the 
necessity of any motive power, and the speed is controlled by a 
brake. Examples of this form are found iu some mines and 
stone quarries, and in apparatus for loading vessels, etc. (See 
Chap. XXII.) Power-driven cable railways for passenger ser- 
vice on inclines are sometimes made with two cables, one for 
driving, and a second for guiding and as an additional security, 
an example being the old road up the Kahlenberg at Vienna. 
When round ropes are used it is desirable to have the drums 
made with spiral grooves, in order to reduce the wear on the 



rr> 




T^ 




Fig. 793. 

rope. The travel on the drum causes the angle of the rope be- 
tween W and L to vary, and to prevent this the device shown 
in Fig. 793 has been used by Riggenbach on the cable incline at 
Lucerne ; two forms being given. The guide sheaves are trav- 
ersed by screw motion, the rope being led off iu a plane parallel 
to the axis of the drum, and in the second form two guide 
sheaves are used for a double cable. 

3. Driving. — This application of tension organs is most ex- 
tensive. The principal forms are given in Fig. 794. The cap- 




Fig. 794. 

Stan a consists of a hollowed drum, the surface of which is 
composed of numerous ribs and the rope is given several turns 
about it. The axial travel produced by the spiral path causes 
the rope to climb upon the larger diameter, from which it is 
easily forced back to the middle from time to time by hand. 
At 4 is a sprocket wheel with Y-shaped sprockets, much used in 
many modifications ; c is Fowler's drum, a form of grip drum 
which grasps the rope automatical]}', and which is discussed 
more fully hereafter. At d is a simple rope pulley, partly en- 
circled by a tension organ under such load as will produce suf- 
ficient friction to prevent slippage ; t' is a chain wheel with 
teeth to prevent the slipping of the links. In all iive cases the 
wheel may drive or be driven by the tension organ. 

By combination of driving and guiding devices many useful 
transmissions are made. 





Several forms are given in Fig. 795 : a is David's Capstan, 
with conical windlass, with a ring-shaped guide roller which 
constantly leads the rope from its travel toward the base of the 
cone. At 5 is a counter-sheave device, the main sheave 7" being 
made with two grooves and the counter-sheave set at a corre- 
sponding angle. This gives increased rope contact, which may 
be multiplied still more by increasing the number of grooves. 
The counter-sheave may also form tlie second pullej' of the 
combination, as at c\ this is used in rope transmission devices. 

Driving tension devices are often capable of being used to 



174 



THE CONSTRUCTOR. 



greater advantage than winding devices, since the direction of 
motion need not be changed and is not limited. For these 
reasons driving combinations are frequently used instead of 
drums, as in hoisting machinery. Chain sheaves with pockets 
to receive the ordinary oval link chain are here applied (see 
I -75). or with flat link chain the sheave engages with the pins 
of the chain. 





Fig. 796. 

Other driving sj'stems are shown in Fig. 796. At a is a double 
lift with water counter-weight. T'is a pulley for round or flat 
belt; the weights O, and Q., are nearly equal, so that a semi- 
circle of contact is .sufficient to j^revent slipping at Zj and the 
friction of contact is sufBcient. 

A reference to the Riggenbach cable road gear, Fig. 7S6, will 
show a similarity to this device, but in Fig. 7S6 a braking de- 
vice is provided at Oj and 0,_ to protect from accident in case of 
breakage of the cable. A similar device, using strains at T, has 
been applied by Green for operating the sluices of the Great 
Western Canal. A*" b js shown the grip-wheel, which has also 
been used for cab'^e driving. In this form the loads may be 
quite unequal without apprehension of a deep groove cutting 
in the drum. Koppen's system is shown at c\ this uses a round 
or flat belt with tightening pulleys L, L, so that sufficient fric- 
tion can be obtained for any given difference of loads ; this 
avoids the unequal action upon the heavily-loeded side of the 
belt, by producing tension upon the otherwise slack side, and 
might be applied with advantage to the driving system of Fig. 
795 r, requiiing but a single tightening pulley, and subjecting 
the rope to only one kind of bending. 

At d is shown a bucket gear, which combines driving and 
guiding, and is much used for conveying in mills, grain eleva- 
tors, etc. If the difference in weight between the sides is slight, 
the tension organ may be a leather belt, but for heavy service a 
chain is used. This device has been in use from a very early 
period for well buckets, and in modern times in mud dredging 
machines. At e is the Weston differential pulley block, a modi- 
fication of the Chinese windlass. Fig. Jgoa. T^ and Tjare chain 
sheaves fast to each other, producing a differential action due 
to their difference in diameter, the whole forming a substitute 
for the older tackle block gear. Fig. 7S4 ,:. 

The form shown at Fig. 796 d demands further consideration, 
as it can be given a series of most important applications. 

,If the tension organ is made a band and placed in a horizon- 
tal or nearly horizontal position, it can be used to convey finely 
divided material simply poured upon its upper surface. Exam- 
ples of this are found in the transportation of grain, also in the 
movement of paper pulp, and many other such purposes ; also 
for convej'ing straw upon chain lattice conveyors, etc. In all of 
these cases the material is kept on the conveyor simply by 
gravity. This condition may be avoided and the capacity ex- 
tended by using a pair of belts, the material to be conveyed 
being carried between them. .\ very important application of this 
principle is found in power printing presses, the delivery of the 
sheets being effected by S3'stems of tapes and bands with great 
speed and accuracy. Band conveyors are also used in needle 
machinery and in match making machines, and many similar 
situations. 

An important application of driving gear is found in the con- 
struction of inclined haulage sj'stems for mine ramps. 




Fig. 797. 

In Fig. 797 is shown the inclined cable system of the Rhenish 
Railway. The driving wind T L, operated by a steam engine, 
works the descending cable on one track and the ascending 



cable on the other. At L' is a tension pulley to take up the 
slack cable and maintain a proper tension. The trains Q^ and 
ft, are connected to the brake cars i?, and i?,, which are extra 
heavy and control the rate of descent by proper brakes. 

In the anthracite coal region of Pennsylvania haulage systems 
are in extensive use for the transportation of coal, some being 
constructed with iron bands, but most of them using ropes. 
The arrangement will be understood from the diagrams in Fig. 
798 and 799, which, with the accompanying data, have been ob- 
tained by the author from their engineer and constructor, the 
late Mr. W. Lorenz. 




The car in which the coal is hauled is not attached directly to 
the cable, but is driven by a dummy D, which is permanently 
connected to the cable. This dummy runs on a narrow gauge 
track, and at the foot of the incline the narrow track continues 
on, so that the dummy /? can go below the main track, as shown 
in Fig. 799, and ou the ascent it can thus be drawn up behind 




Fig. 799- 

the cars which have been placed by the shifting locomotive 
The steam engine and drawing gear is placed at the head of the 
incline, as shown in Fig. 798, and the cable is led, as shown by 
the arrows, that it passes twice over the driving wheel T, each 
time covering about J4 of its circumference. The dummy cars 
/), and D., are connected by a secondary cable passing over the 
tension sheave L' ; this secondary cable maintains the proper 
tension on the main cable, whether the load is at the head or 
foot of the incline, or on the horizontal. The tension car. is 
given a play of 75 feet to provide for the necessary variation. 

A different form of cable haulage is found in the .system in 
use between Liittich and Ans, and sketched in Fig. 800.* 




Fig. Soo. 

In this case the incline is divided into two sections, which 
make an angle with each other as shown ou the plan, and be- 
tween which is a short level space. On this space is placed the 
steam engine and driving wheels Ti, 7",, T^, T^, each wheel 
having its own engine, two engines always diiving and two 
being at rest ; L' are the tension sheaves. 

In this, as in the preceding case, it wiil be noticed that the 
cable runs continuously in the same direction, differing in this 
respect froin the previously described winding and reversing 
system. The cable is brought to rest in transferring the cars 
from one plane to the other in order that this may be readily 
and convenientl}' done, but should this be avoided by running 
them over the connection, by momentum or otherwise, the ad- 
vantage and usefulness of the system would be greatl)' increased. 

This has been done in the cable tramways of Halliday and 
Eppelsheimer, first used in San Francisco, .and shown in dia- 



^ See Weber'.s " Poitfolio John CockeriU." 



THE CONSTRUCTOR. 



175 



gram in Fig. Soi. This is most effectively applied on the trolley 
streets of the city, for which it is admirablj' adapted. 




Fig. Soi. 

The endless cable runs in an iron way between and beneath 
the tracks, the power being at T and guide sheaves at L, L, 
with suitable driving and tension mechanism. The cars grasp 
the cable by a gripping device through a narrow slot in the 
trackway. The guide sheaves at the bases of the inclines and 
sides of the curves permit the grip to pass, and when the foot 
of the hill at the end of the road is reached, the grip is released 
and the car transferred to the other track as at M\, and in sim- 
ilar manner shifted at the other end, IV^_. The weight of the 
cars on the down grades counterbalances those on the up 
grades, and so the motive power has only to overcome the fric- 
tional resistance. The cable system of tramways has been ex- 
tended to Chicago and manj' other American cities ; also in 
London, and a cable system of canal towage has been projected 
by Schmick for the proposed Strasburg-Germersheim Canal. 

When it is practicable to propel the cars by a .suspended cable 
from overhead a different arrangement maj' be adopted. 




Fig. S02. 

Fig. S02 is a diagram of a system operated by a suspended 
chain. The descending cars Q^ are loaded and the ascending 
ones C?., are empt)^ and the speed is controlled by a brake at B. 
If the action is in the reverse direction, a driving engine must 
be applied at T. A similar arrangement is much used in coal 
mines which are entered by inclines. The chain is attached to 
a fork on the cars. 

The system of overhead cable tramway, which has been 
brought to a high state of efficiency by Bleichert, is based on 
the same principle as the preceding, but for much lighter loads. 
The sj'stem consists of a cable tramway in which a stationary 
cable is substituted for the trackway. The running cable is 
commonly called the pulling rope, and runs underneath the 
stationary rope. The cars consist of a combination of grooved 
sheaves, from which the bucket or other receptacle is suspended 
by curved arms. The stationary cable is supported upon round 
poles, and the arrangement of the stations is shown in the dia- 
grams of Figs. So3« and 803^. 




Fig. 80317. 

The stationary cable connects with the suspended tramway at 
Sr S// and Sifll S'V. At So is the anchor of the stationary 



counter-sheave, as in Fig. 795^, to obtain increased tractive 
power. 

Fig. 804 shows a plan view of a double system. 



pa 



fj 



SL. 



C/^// i,/,/jJu//u/^,„J//J^-^. 





Fig. %oib 



cable, with a tension weight at Z,. The driving sheave is at TJ 
driven by connections to the engine at A', and at L' is the ten- 
sion device for the pulling cable. If the service is heavy the 
cable is carried twice around the driving sheave 7, using a 




Fig. S04. 

At A'l is the motive power for systems / and //, and at K^ the 
motor for system ///. The driving sheaves are at 7, the coun- 
ter-sheaves at G, and the tension sheaves at L' . 

The supporting columns for the stationary cable must be 
stiff, and often quite high. 

a. b. 




Fig. 805 shows the forms used b}' Bleichert, a being used up 
to 24 feet high, b for heights between 24 and So feet.* 

In Fig. 806 is shown a combination of driving and guiding 
systems in which the guiding and driving sheaves are combined 
upon the car Q, and the tension organ is fastened at two points 
So So on the path of the car Q. 




SflSo 






Fig. S06. 

The motive power is on the car and operates the shea-^ e 7. 
In the form shown at a, a Fowler grip sheave is used at T, this 
form being suitable for a rope system, while the form shown ai 
b is better adapted to be used with chain. 

The system shown in Fig. 806 5 is also adapted for hauling 
boats, and has been used by Harturch for operating the railway 
ferry across the Rhine at Rhinehausen. The ferry baat in this 
case is guided by a stationary cable securely anchored, as in 
Fig. S07, the anchorage being up the stream, and the force of 



T ^ T 



Fig. 807. 

the current keeping the cables taut. The equilibrium of these 
forces enables this to act in the same manner as the stationary 
cable of the Bleichert .sy.stem, the difference only being that the 
load, instead of being suspended from the cable, exerts a lateral 
stress. The driving cable is similar to Fig. S06 b, and is beneath 
the surface of the water. 

If we imagine, in the combination of Fig. 806, that the 
traveling vehicle O may be longer than the distance S^ S^y 
which is the full length of the tension organ, the principle w'ill 
not be altered, but the action will be modified, since the rela- 
tions of the traveling vehicle and the tension organ are now 
inverted. The ends of the tension , organ can now be joined 




* On the tramway at Liker-Vashegy, poles of 140 feet high are used. 



1/6 



777^ CONSTRUCTOR. 



together, or in other words it can be made endless, and if heavy 
enough, its weight can be caused to produce enough friction on 
the bed of the stream to furnish the necessary resistance. This 
is the construction of Heuberger's chain propeller, Pig. So8, as 
improved by Zede 



L L ^^. L L r L 

^— o^^-o — o — o — ^ 




Fig. 808. 

T is the driving sheave for the chain, L, L, L are guide 
sheaves, Zj is a movable sheave to take up a portion of the 
slack chain when passing into shallow water. The system is 
made double, being placed on each side of the boat, and each 
side is driven independently, so that sharp curves can be turned.* 
If, in the case of a tension organ driven by a revolving pulley, 
there is not sufficient tension given, the friction becomes insutS- 
cient to overcome the resistance of the load ; if the necessary 
tension is externally supplied and removed periodically, a con- 
tinuousl}' revolving piilley can be caused to produce a lifting 
and dropping action of a given load. This plan has been 
a'^opted in some forms of drop-hammers, of which P'ig. S09 is 

the arrangement. 7" is a pul- 
ley running continuously in 
the direction of the arrow, O 
is the drop weight, //a handle 
by which the operator applies 
and releases the tension which 
causes the pulley to drive or 
slip. 

The applications of running 
tension organs which have been 
thus far considered, are +nose 
in^which the device has been 
used either to lift weights or to 
transport the same from place 
to place. One of the most 
important applications, how- 
ever, is that of transmitting 
rotative motion from pulley to 
pulley, an operation which can 
be almost indefinitely repeated. 
This combination includes all numerous forms of belt, rope and 
chain transmission. Fig. Sio. The necessary tension for this 
purpose is sustained by the journals and bearings of the pulleys, 
also being modified by supporting or by tightening pulleys. 
The two portions of the tension organ are distinguished as the 
tight and slack sides respectively," and many modifications of 
this form of transmission are discussed more fully hereafter, 
(see Chap. XX to XXII ). 

There is one application, however, which is appropriately 
discussed in this place, namely, that in which rotative trans- 
mission between pulleys upon stationary axes is combined with 
pulleys upon a movable member, thvis enabling motion to be 
transmitted from a stationary source to a moving body, Fig. Sli. 




Fig. S09. 



Fig. Sio. 




Fig. 811. 

In case a, one of the driven pulleys is mounted upon a car- 
riage, saddle, trolley, or the like, and may be shifted in posi- 
tion upon its wa3's or track ; the tension is sustained by the 
three guide sheaves. Applications of this form, using belting, 
are used upon planing machines by Sellers, Ducommun & Du- 
bied and others. With rope driving gear it is used to operate 
the spindles upon the carriage of the self-acting mule, also for 
operating traveling cranes by Ramsbottom, by Tangye, and by 
Towne ; being combined by the latter with the squaring device 



* The following data of performance are given by Zede : Capacity, 500 tons ; 
length over all, 230 ft. ; breadth, 21^$ ft. ; depth. 6^ ft ; midship draught, 3iJ4 
in. The chains were of cast iron, weighing 275 pounds per yard" two en- 
gines cf 150 I. H. P. gave a speed of 3.72 miles (!) per hour. 



as shown in Fig. 784 f, and effecting all the functiotis of the 
crane, including bridge and trolley travel, as well as the hoist- 
ing and lowering of the load. 

The form of Fig. '&\\b diifers from a in that both sides of the 
belt or rope are used to transmit power. .The stationary pulleys 
7"j and T^ here drive the movable pulleys T,_ and T^. These 
driven axes can be utilized in various manners, as, for example, 
to operate a windlass device for the propulsion of the carriage 
O ; an example of which is found in Agudio's cable locomotive, f 
In this device the pulle}'S T^ and T^ drove a friction train which 
operated a drum connected with a stationary cable as in Fig. 806. 

A more recent device is shown in a modification of Fig. 811 a, 
as shown in Fig. 812.J 




W^vLj/L^ ii ^»ff^^M( 4h if^M,^ f. // - ^-.^ 



Fig. 812. 



This construction, which is in use at the Soperga-Rampe at 
Turin, consists of a double rack, placed between the rails as 
shown at b, which also shows the gearing by which car is 
driven. The motive power is placed at the foot of the incline 
at T, G, the 500-horse power engine running continuously in 
one direction. The cable is carried upon the overhead guide 
sheaves Z, and passes around the pulley Z,, and through the 
sheave system T T' of the locomotive, and is supported also on 
guide sheaves under the track, a tension pulley being placed at 
L'. The velocity of the driving cable is four times that of the 
cars, and the descent is effected by gravity alone under control 
of a brake. During the descent the bevel gears on the shaft of 
the driving pulley are released by friction clutches at K, thus 
rendering the car independent of the cable. 

The foregoing condensed description is nevertheless fully 
sufficient to indicate the extreme service of which tension organs 
arc capable in machine design. No less than seven sj'Stems 
have been shown for railway use, and four for boats. This is 
the more significant since it will be remembered that cable pro- 
pulsion had been abandoned for railway use, but yet appears to 
now be revived with increasing success. 

Our division into Guiding, Winding, and Driving systems 
enables different devices to be placed in corresponding classes. 
There yet remains to be considered the co-existing action of 
many of the devices, such as pulleys, windlasses, cranes, etc., 
in which a negative motion may be given to the tension organ 
b}^ the descent of the load O under the action of gravity. \ 
This action can be fully determined by reversing the previously 
considered movement for the backward motion. In the com- 
mon belt transmission. Fig. Sio, the action is reversible, as is 
also the case with the simple pulley. Fig. 794 fi'- 

The case is different, however, with the rope tackle Fig. 784c 
and the differential block Fig. 796(', which are therefore here 
considered in the more general form of Fig. S13. 

If in these forms the 
cord Z is pulled in either 
direction the lower 
sheave will be also 
moved up or down pro- 
portionally. At the pre- 
sent time systems using 
endless cords are under 
? I [,1 f i f I consideration, but fre- 

VJ.V 2 \,^ „ VJ^^ 2 qusutly choice is to be 

/^l"^ /l^ made as to which por- 

i I X ( J, « tion is best used. It will 

be seen that the system 
of Fig. 806, which is 
made with both ends of 
the cable secured, can 
also be considered as a 
portion of an endless 
system similar to Fig. 
80S, and other endless 
systems are found in 
Fig. 7S4 d and e ; also 
Fig. 813 b, which differs 
from a only in the run- 
ning of the rope, the united ends being marked by a cross. If 




L 





Fig 



I See Thomas Agudio. Memoire sur la Locomotive funiculaire, Turin, 1S63. 
tSee Bulletin de la Soc. d'Kucouragenient, Vol. XVI., 1869, p. 48. 
I Kinematic force closure. First discussed in the Author's Theoretical 
Kinematics, p. 575. 



THE CONSTRUCTOR. 



177 



If we bring the applications of Figs. S06 and Si i into a general 
form in which the path of travel shall return upon itself, we 
have Fig. S14 a. If the guide sheaves are removed and the 





1 1 



Fig. S14. 

cord crossed, the simpler form of Fig. Si4 b is obtained. The 
rotation of the pulley 7", causes travel around the stationary 
pulley T^. The old form of Agudio's cable locomotive may be 
represented by a similar diagram, Fig. S14 c. The shaded pulley 
T.^ is held stationary, while the concentric pulley Zj, is assumed 
to revolve ; this causes the system to revolve in a circular path, 
the whole forming a differential or epicyclic system. 

Fiualh' it may be remarked that in electric transmission 
systems a similar analogy exists to the above combinations of 
tension organs of wire and cable in various forms. 



I 263. 
Technologicai, Applications of Tension Organs. 

In addition to the preceding applications of tension organs, 
they are also used in numerous forms of machine tools, ;'. e., as 
organs for the alteration of the form of bodies. 

A straight blade of steel furnished with teeth forms the well- 
known frame or gang saw used in numerous wood-working 
machines. When made without teeth, and used with sand and 
water, it becomes a stone-cutting saw, or in the form of a wire 
charged with oil and emery or diamond dust, a saw for the 
hardest materials, in which case a high tension must be given 
to the wire to prevent lateral displacement. The saw blade may 
be given a vibrating motion in a device such as Fig. 7S9 b for 
use as a scroll saw. In all these cases a reciprocating motion 
is used. Tension organs are also used as running members for 
sawing, the form of Fig. Sio becoming the well-known band 
saw. Very fine band saws have been made, and also saws of 
wire, these having been used as long ago as 1S77 by the writer, 
suggested by the saws used for precious stones. 

An ingenious form of wire saw has been made by Zervas for 
cutting blocks of lava or stone from the original bed, as shown 
in the diagram Fig. 815. 




Two small shafts are sunk in the stone, and the guide pulleys 
inserted as shown, the endless wire being fed down by the 
screws. The cutting is effected by using water and sand, and 
the cord is formed of three twisted wires, although more re- 
cently a single smooth wire, with twisted one wound above it, 
has been used, the outside diameters being %" to -f/'- 

A patent was taken out in Germany b}' Paulin Gay in 18S2 
for an apparatus for cutting a block of stone into slabs by the 
use of a number of wire saws. 

Polishing belts are another example of tension organs used as 
tools, the flat side of the belt being used, impregnated with 
polishing material. Such belts, used in the nickel-plating 
establishment of Neumann, Schwartz & Weil, at Freiberg in 
Breisgau, are operated at a speed of over 6500 feet per minute. 

Tension organs are of frequent use in many details of spin- 
ning machinery, acting both for guiding and winding ; also in 
numerous other forms of textile machinery. 

Chains are especially useful for dredging machinery, working 
in wet or dry material, also for handling coal. 

In musical instruments we find tension organs of definite 
dimension and stress, as sound producing machines. 



§264. 
Cord Friction. 



When a tension organ which is loaded at both ends is passed 
over a curved surface, there is produced between the tension 
organ and the surface a very considerable sliding friction. 
Since this friction will first be mathematically considered in 
connection with the subject of cords, it will be given the gen- 
eral name of cord friction. The curved surface over which the 
cord is passed is the pulley, and the motioii of the cord takes 
place in the plane of the pulley. If the 'ension T on the 
driving side of the cord is to overcome the cord friction F, as 
well as the tension i of the driven side, we have for the value 
of the friction, F=s T— i. It is dependent upon the magni- 
tude of the angle of contact a and upon the coefficient of fric- 
tion/, but is independent of the radius 7? of the pulley ; it is 
also dependent upon the influence of centrifugal force. For 
these conditions we have : 



r=;?tf/»(i-z) 



(237) 
(23S) 



In these e is the base of the natural system of logarithms := 



2.71S2S, and .:: = 12 



gS 



V being the velocity of the tension 



organ in feet per second, S the stress in its cross section, y the 
weight of a cubic inch of the material, and g the acceleration 
of gravity = 32.2. 

Example. -In the capstan shown in Fig. 794rt,let/= 0.21, a = 6 tt- = 3 con- 
volutions, z = o. We then have / a = 0.21 X 6 X 3. 14 = 3-958, say 4, and 
F = t {2.718^ — A = t {54-6 — = 53 6 t. This shows the friction npon ti;e 
capstan drum to be nearly 54 times the puU upon the free end of the cord. 

The influence of centrifugal force becomes important at high 
speeds, and when the tension organ is under small stress. For 
hemp or cotton rope, or for leather belting, we may take y = 
0.035, and for wire rope about nine times so great. 



The value of 5 in the formula ; 






is properly con- 



sidered a function of ; 
stant value for the arc 1 
for the values of i — z 



and we may therefore assume a con- 
and thus calculate the following table 



TABLE. 



5-. 


Value of Coetlicient i — z for Centrifugal Force. 


s. 




I'elacity of Rope in Feet per Second, 


















Rope. 


20 


40 


60 


80 


100 


Rope. 


400 lbs. 


0.987 


0.948 


0.882 


0.791 


0.674 


3,600 lbs 


600 " 


0.991 


0.965 


0.922 


0.861 


0-783 


5,400 " 


800 " 


0.993 


0.974 


0.941 


0.896 


0.837 


7,200 " 


1000 " 


0.995 


0.980 


0-9.S3 


0.916 


0.870 


9,000 " 


1200 " 


0.996 


0.982 


0.961 


0.930 


0.S92 


10,800 " 


1400 " 


0.996 


O.9S5 


0.966 


0.940 


0.907 


12,600 " 



This table serves both for hemp and for wire rope by taking 
the ninefold value of.? in the right hand column for wire rope. 
It should be observed that the velocities are in feet per second. 
It will be seen that for high speeds a high stress in the tension 
organ is necessary in order to oppose the action of the centri- 
fugal force. 

In order to simplify practical calculations we may substitute 
for the e-xponent/a (1 — z^ in each case the form/'' a ; that is, 
instead of using the actual coefficient of friction _/, taking an- 
other oney, which is equal to (i — z)f. If it is a transmission 
system, as Fig. Sio, which is under consideration, the friction of 
the cord, belt, chain, etc., must at least equal the transmitted 
force P, hence also must the stress be that of a cord friction 
^ /", which gives for a minimum value of T: 



T 



efo- 



p — i 



(239) 



whence 



^ ^ p = e/'a 



Both or' these values are absolute numbers. The ratio 



(240} 
7 



"P 



indicates the amount of stress which must be given to the ten- 



178 



THE CONSTRUCTOR. 



sion organ, and hence may be called the stress modulus, and is 

T 

designated as t. The ratio — , we may, in like manner, call the 



modulus of cord friction, and indicate as /; 
for both are given in the following table. 



A series of values 
Moduli for Coid Friction and Stress. 




Exaiiipie — Krc o^ contact = tt ; coefficient of friction^= oi6, velocity z' = 80 
feet. The teiisi on orgran is a leatlier belt under stress of 400 lbs. per square iuch. 
We have from the first table i — r = 0.791, hencey' a = 0.791 X 0.16 tt = 3 976, 
or nearly o 4. From the second table this gives p = 1.49 and t = 3.03, that is' 
over three tin-es the above stress on the belt would be required to overcome 
the frictional resistance. If z/ = 20 ft., the value i — 2- = 0.987, and /' a = 
0.496 or about o 5, and the modulus of stress t = 2.54. 

In order to make these relations more apparent, they are 
shown graphically in the diagram, Fig. 816, in which the 'scale 
upon the upper horizontal linegives the values for both moduli, 
wliile the vertical scale on the left gives corresponding values 
01 the product y"'' n. 




The superficial pressure/ of the tension organ upon the cir- 
cumference of the pulley increases as the belt or cord passes 

from the slack to the tight side. It is equal to ~-^^, in which 

b' is the breadth of the surface of contact of the belt. Now for 
any cross section q, the force Q = q S. Hence we have : 

/ Q 



S 



b' 1< 



(^41) 



from which it will be seen that the pressure p can easily be kept 
with moderate limits. 

Special applications of this formula, aud of the diagram, Fig. 
S16, will be given hereafter. 

'i 265. 
RoPE.s OF Organic Fibres. 
Hemp Rope. — The form in most general use is a round hemp 
rope twisted of three strands. This is twisted "loose" or 
"tight," according as it is desired to be 
more or less flexible. The cross section 
of a three-strand round rope, in which rf 
is the diameter of a single strand, is 

3 — ('', whence <! bears the following 

relation to the diameter d of the circum- 



scribing circle : d 
= 2.15 6, Fig. Si 7 
cross section q 



.6 (1 + -^- 
V COS 30 

This gives for the 
d'-. On account 



6.16 

of the spiral twisting of the strands, and 
their compression upon each other, this 




Fig. Si 7. 
that is about oS times the value of 



may be taken q =^ — 

the ftill cross section. Good hemp rope, when loosely twisted, 
will bear a stress of 1700 pounds, and when tightly twisted, 
about \}4 times as much. For convenience of calculation wo 
may assume the cross section to that of the full circle d, if, in- 
stead of the full stress, we take onlj' fas much, or 1400 lbs., 
and 2100 lbs. We then have for the force P, for: 

looseh' twisted rope rf = 0.0^ v v°; and /'= iiii rf'M , , 

/— y ■ (242 

tightly " ■ " rf = 0.024 V/'; " P=ri67Td^' 

The radius J? of the pulley should never be less than 3 to 4 rf 
for loosely twisted rope, and not less than 6 to S rf for tightly 
twisted rope, the diameter being measured to the centre of the 
rope. I"or heavy service, as for hoisting machines, J? should be 
not less than 2; d. 

Flat hemp ropes are made by sewing 4 to 6 round ropes to- 
gether, each, rope being then proportioned to bear j or ^ the 
whole load. 

The running weight G^ per foot is as follows : 

For loosely twisted rope, G^ = 0.325 rf' I 

For tightly " '• G„ = 0.467 d' \ .... (243) 

and approximately for both /'= 3400 G^ ) 

The latter assumption is based on the same number of fibres 
in both cases. The following table gives values for three-strand 
hemp rope. 



1/, 

H 
I 

2 

2 'A 



Loose Twist. 


Hard Twist. 


J°- . 


Co. 


f. 


Co. 


275 


o.oSi 


397 


O.I16 , 


621 


0.183 


893 


0.263 


967 


0.284 


1,-89 


O.40S 


II05 


0.325 


1,588 


0.467 


1726 


0508 


2,481 


0.729 


2485 


0.731 


3,572 


1.050 


4420 


1.300 


6,351 


I.86S 


6906 


2.031 


9,923 


2.919 


9945 


2.925 


14,290 


4.203 



Fig. S16. 



According to (243) a rope L feet in length, hanging vertically. 

is loaded L of its working strength already by its own 

3400 =" " 

weight. If Z =- 3400, the entire practical load would already 



THE CONSTRUCTOR. 



^79 



be applied, and this may be cousidered practical working 
lengtli of the rope. We have for the available practical work- 
ing load : P' A ^~ L P= Par P' = P ( i ^~ L 

3400 V 340° 

V A vertically suspended rope will break by its own weight 
■when its length reaches about 2000 feet, since the modulus of 
rupture is about S500 lbs. for loosely twisted rope, and about 
14,000 lbs. for tightly twisted rope. The above length (2000 ft.) 
may be called the length of rupture. For a cord suspended 
in the water, as for deep sea .sounding, the leugth of rupture is 
about twice as great. For very heavy stresses three simple 
strands are insufficient, and the strands themselves are each 
made of smaller strands, as in cable construction. Very heavy 
cables are also nia,.'e of more than three strands. 

Cotton Rope. — Cotton rope has been used of late for purposes 
of transmission, and is usually made with three strands, very 
loosely twisted. It opposes a resistance to rupture of about 
7500 pounds, reckoning the full sectional area, and is operated 
under stresses ranging from 1000 to 2000 pounds. It is used for 
driving spindles in spinning frames and mules, and in the snail 
drum movement, as in Fig. 7S7,* and is also used for operating 
traveling cranes on the Ramsbottom system. 

Driving ropes are usually operated over grooved pullej'S, the 
radius of the semicircular groove being slightly greater than 
that of the rope. In machine construction the sheaves are 
usually of cast iron, and in ship's tackle they are made of 
lignum vitse. 

The sheaves revolve on cylindrical journals, and recently 
roller bearings are being used, Fig. SiS.f 



)fhf^/ 






Fig. 818. 



Fig. S19. 



When the pressure is moderate, the rollers may be made of 
hard bronze, but for high pressures the rollers, riug and journal 
■should all be made of hardened steel. In case of e.Mremely 
high pressures bronze bearings with metaline may be used, the 
Tnetaline being a solid lubricant imbedded in recesses in the 
box, Fig. 819.1 Such bearings were used most successfully in 
the construction of the East River Bridge at New York, oper- 
ating for an entire year without requiring lubrication. 

1266. 

Wire Rope. 

Wire rope is usually round, and made with 36 wires, since six 
-strands are used, each containing six wires. Each strand con- 
tains a small hemp core, and the strands are twisted about a 
central core of hemp. These hempen portions are of greatest 
importance in the construction of wire rope for transmission 
(see ^ 26S), and should be made of the best material. For sta- 





FiG. 820. 

lionary ropes the hempen strands may be replaced by wire, 
giving 42 or 49 wires, and proportionally increasing the strength 



* 111 a spinning mule of 844 spindles, by J. J. 
thure, a rope 22 mm. ^o.S66 iu.) operates under a 
taking the full cross section. 

t Martini's design, used in the Italian navy. 

X John Wallace & Co., London ; Selig in Berlin. 



Rieter & Co.. of Winter- 
stress of 1.6 kg. (2275 lb.), 



of the rope. The strands of six wires may be combined to 
make ropes of 48, 54, 60, 66, 72 wires, etc., and other combina- 
tions are also used. 

In Fig. 820 is shown at a a section of a rope of 36 wires, and 
at /; a difft- rent form of 60 wires, both being made with 2ores of 
hemp for the strands as well as for the ropes. For the external 
diameter of the wire ropes of the preceding form, when the 
wires lie iu close contact, we have : 



i — 36 
T 



48 54 60 
= 8.00 10.25 11.33 12. So 



66 72 1 
13.2S 14.20 



(244) 



in which / is the number of wires, d the diameter of a single 
wire, and d the diameter of the rope. 

In some later kinds of rope they do not lie in contact with 
each other, but are separated slightly by the hemp, in which 
case the diameter will exceed the previous figures by 10 to 15 
per cent., but after a period of use the diameter becomes reduced 
to the sizes given above. When the strands are made without 
hemp cores they are arranged in the following manner : \ 

3 7 10 14 16 19 

while with hemp cores the numbers are 

56 7 S 9 10. 

The number of strands runs from 3 up to 4, 5, 6, which latter 
is most used, and on up to 7, 8, 12, 14, 16, 19. For cables which 
arc required to resist heavy stresses and also to possess great 
flexibilit}', the same construction is employed as for hempen 
cables, the strands themselves being composed of twisted ropes ; 
the number of strands is 3, 4, 5 or 6. Flat cables are also made 
of a number of parallel ropes. The number of ropes is 4, 6 or 
8 ; the number of strands in each rope 4 to 6. 

Example. — K heavy cable of steel wire is made of 6 ropes, each rope of 19 
strands, each strand containing 7 wires. The total number of ^vires = 
6 X 19 X 7 = 79S. Diameter of wire 6 = 0.055". 

Well made rope is so wound that the load produces a uniform 
stress upon all the wires, so that, when i = the number of wires, 
/'the load, 6' the stress on the wire, we have 



P=Si 



|S2 



(245/ 



The diameter of wire varies from 0.04" to o. 14". If the rope 
is required to be very flexible the wires should not be more than 
o.i" in diameter. 

In the passage of the rope over a sheave or pulley, of a ladius 
R, the individual wires are subjected to bending, which, under 
the action of tension and compression l,see ^ 8), produces a 

stress of a magnitude i = tt^ in which E is the modulus of 



elasticity of the material, 
taken at := 28,440,000. 



This gives .9 = 14,220,000 



2/v? 

For steel or iron wire E may be 



(246) 



The stress s, which is produced on the tension side by bend- 
iug, mubt be considered in connection with the stress 5 produced 
by the load P, in order to arrive at the total stress. In order to 
avoid a permanent set, it is necessary that the sum S-{- s should 
not exceed the modulus of elasticity. The actual magnitude of 
R becomes a minimum when s =^ 2 S\ that is, the stress due to 
bending becomes double that due to the working tension. 

Whatever may be the relation between the pulling stress S, 
and the bending .stress s, the total stress on the material will be 
the sum 6" -\- s. 

If it is desired to consider the security against rupture as well 
as the possible overstepping of the elastic limit, the value of 
6"+ 5 must be taken into account. The Prussian Government 
rule places the modulus of rupture A' if bteel wire at 163000 

pounds, or with a factor of safet}' of 6, the stress 5'^ — ~- — 

= 27,166. If we take the rase of a rope of 42 wire.'i, its diame- 
ter rf ^- ID'S, and making ttie pulley diameter = 75 </, we get 
R = 37.5 d. This gives the bending stress, according to (246J, 



5 = 



14,220,000 S 



37-5 X lod 
The sum 5"-|- .f 



= 37,920. 



actual factor of safet)' of 



7,166 -f- 37,920 ^ 65,086. 
63000 



This gives au 



650S6 



= 2.5. 



§ In American mining: machinery, six strand ropes of 19 wires, with, hemp 
cores in the middle, are much. used. 



I So 



THE CONSTRUCTOR. 



The relation of the 
stresses in the various 
parts of the rope are 
shown in Fig. S21. 

On the right, the 
tension side, there is 
the tension stress ( -|- S) 
and the bending stress 
(+ s), giving a total 
of 5 + .f. On the left 
thetensionstress(+ S) 
is diminished by the 
reverse bending stress 
(— s). The neutral 
axis: is therefore shifted 
from the middle at 
N, to a point toward 
the concave side of 
the bent rope at jV. 

Wire rope may be made either of iron or steel wire, and its 
fabrication has greatly advanced within recent years. The fol- 
lowing data are applicable to the various grades :* 




+ ® 



Fig. S21. 



i^Iaterial. Elastic Limit. 

Annealed Iron Wire . . 42,000 

Bright Iron Wire . . . 56,000 

Steel Wire 64,000 

Steel Wire 78,000 

Steel Wire 99,000 

Steel Wire iij.ooo 

Steel Wire 142,000 



Modulus of Rupture 

56,000 

So, 000 

85,000 
142,000 
170,000 
213,000 
256,000 



It will be evident that no general rule can be given as to 
material, but that definite figures should be obtained for the 
material to be used in each case. For high speed rope the wire 
should be both smooth and strong, with a modulus of rupture 
of about 170,000 lbs. If we then take a working stress S = 
28,000 lbs., and a bending stress .? ^ 28,000 lbs., we have 5+5 
= 56,000 lbs., which gives about threefold security.! 

14 220 000 

For i = 28,000 we have R = ^ <! = 500 (5. If R is 

28,000 

made less, the security will be reduced ; if greater, it increases, j 
The durability of the rope for mining servtce is increased by 
galvanizing the wire. 

For standing rigging of vessels galvanized annealed iron wire, 
with a value A'= 56,000 is used, while for running rigging steel 
wire rope (A'= 170,000) is being more extensively used, this 
also beuig galvanized. The latter rope is also suitable for 
cables. Hawsers are frequently made from iron wire, with a 
modulus of rupture K ■= 56,000 to 70,000. The cables for steam 
plowing machinery should be made of the strongest steel wire, 
K ^ 256,000. 

Wire Cables for power transmission are discussed in Chapter 
XXI. 

The cables for suspension bridges are not made from twisted 
strands, but the wires are laid parallel and held in position by 
bands of wire every two or three feet. § 



* See the researches of J. \V. Cloud on steel wire in connection with the 
Enier>' Testing Machine at the Watertown ArseTial. Trans. Am. Soc. Mech. 
Eng'rs, Vol. V. 

t The Prussian rule requires S = -^ K, which gives about 28,000, and R — 


37s 5, which gives i" — 38,000, hence the security is only about 2^, or less than 
given above. 

Prschibram has used with best results, S = 23,000, 5=27,000; also 5" = 
22,750, s = 36,000, but finds that a value i 27.000 to 28,000 lbs. is better for the 
preservation of the rope. (See j; 26S.I In considering the question of pullev 
diameter, the ratio to the diameter 5 of the wire should be taken, not tha't 
to the diameter d oi the rope. 
R 

t If -J- is made so small that S + J is greater than the elastic limit, the 

rope will receive a permanent set. 
This, however, is uot always dan- 
gerous. 

In Fig. 822 the curvature i . i 
may produce a stress upon the 
concave side of the wires which, 
■when added to 5, may not exceed 
the elastic limit. If. however, a 
reverse cnr\'ature be given, as at 
3.3, there may result a set, as 
3' . 3', and too frequent repetition 
of this reversal may become dan- 
gerous. This is shown in the case 
of hoisting drums, such as Fig. 
792 c, in which the rope ll\ L«, 
■which is subjected to reverse bend- 
ing, has been found to last only 
about ^^ as long as the rope U\ Zj. 

fi .'^mong important suspension 
■bridges are those built by Roebling in America, notably the Niagara, snd 
llie.East River bridges. 




I 2 3' 3 

Fig. 822. 



? 267. 

Weight of Wire Rope axd its Infeuence. 
A rope of parallel iron or steel wires, exclusive of any bands, 

will weigh, per foot, 0.28 ( 12 — i S'' ) , in which i is the num- 



■•)• 



ber of wires and i the diameter of each wire. For twisteri 
rope, the twist and the hemp core increases this value from ijs 
to 1% as much, or an average of !](, times. This gives for the 
running weight per foot 



Go 



1.92 — z (!-' = 3.07 i 6- 



(247) 



This is also true for flat ropes, the value of the coefficient for 
cable ropes being increased as above from i Vs to IJ4, usually 
about lib times. For deep mine hoists the weight Go exercises 
a marked influence upon the section of the rope. If Z /t is the 
length in feet of the vertical hanging rope carrj-ing a load P sX. 

Its end we have : P-{- L Go =^ S — / c!-, whence for ordinary- 
round wire rope : 



P=S- 



/,S^(.-3.92^) 



(248) 



Example i. — Let the depth of shaft L = 1640 ft. Wire rope of steel, A' = 
170,000, i-= 28,000, P= 44oi lbs., and i = 36. 



— X 440= 



^ 0.0056 



which gives I 



28,000 X 36 (i — 0.229) 
: 0.075. \i L ^ O we get 6- =^ 0.0034, and 5 = 0.058. 



The above discussion enables us to determine the length Lt of 
rope which ■would produce by its own weight the stress S in the 
uppermost cross section : 



■■ 0.25 5 



(249) 



This may be called the load-length for the stress 5. Should 
the shaft reach a depth equal to the load-length, no weight could 
be suspended to the rope without exceeding the permissible 
stress ir. If S is equal to the modulus of rupture, the rope 
would be broken by its own weight. This rupture-length may 
be designated bv Lz, and is 



= 0.25 K 



(250 



Example 2. — For round wire rope of uniform cross section the rupture- 
length L= is as below for the corresponding strength : 

K 56,000 80,000 85,000 142,000 

I.Z 14,000 20,000 21,250 35,5°o 



1 70,000 
42,500 



213,000 
23,250 



256 000 
64,000 



For very deep shafts it has been found advantageous to make 
the rope a body of uniform resistance, which would make both 
load-length and rupture-length unlimited. The formnlse for 
this purpose have been already given in § 4. The taper to the 
rope may be given in two different ways. Either a constant 
diameter (J of ■wire, and varying number i, may be used ; or a 
constant number i, and variable diameter 6. If the smaller 
diameter of wire = t'o. or the minimum number of wires = io, 
we have for any depth .i- : 

log T- or log --- = 0.4342945 ;' --;. 
lo Oo .0 

In this )' is the coeflicient of weight which, for round rope, 
we have found to be = 3.92. Substituting this value we get : 



log ^— or log --^ =^ 1.68 -- 

lo Oo ' o 



(2SI) 



£.xa>nj>!c 3.— If the value of ^ be takeu as 28,000, we have for the following 
depths : 

X = 1000 1500 2000 2500 3000 3600 

— 0.036 0.054 0.0714 0.0S9 0.107 0.121 

-7— 1. 115 1.123 i-3^S 1.4.11 1.512 1.597 

lo 

r— 1.072 I.IIO 1.148 1. 187 1.230 I-263 

60 

These values will serve to approximate the intermediate cases Q 



[1 In the Prschibram mines taper ropes are in practical use. The rope ia 
the Adalbert shaft is as follows; j°= 3850 lbs., 01 which 2200 is useful load, 
J^ = 74.S", and the rope is madf in 7 sections of six part strands and eight 
hemp strands. 



THE CONSTRUCTOR. 



i8i 



The great weight of the twisting rope has led to the use of a 
double lift, each half of the rope assisting to counterbalance 
the other half, or another plan is to use a conical drum, to 
equalize the power.* The spiral winding of flat ropes also 
serves to equalize the leverage of the drum, and by a judicious 
selection of drum diameter, this may be very successfully done. 
Flat ropes are little used in France, but are common in Bel- 
gium, and their use is iucreasiug in England and America.f 
Ropes of copper wire are used for lightning conductors, and 
these are also made of iron wire rope with a core of copper. 

I26S. 
Stiffness of Ropes. 
The resistance of stiffness of ropes must be considered both 
in hoisting and in driving ropes. The measure of this resistance 
is the force required to move a rope hanging over a very easy 
running pulley, both ends of the rope bearing the given load Q. 
It will be observed that the winding-up side of the rope does 
not hangas closely to the pulley as does the other side, and that 
the lever arm of the two sides is constantly changing. Ej'tel- 
wein's formula gives for the stiffness ^ of a hemp rope of diam- 
ter f/: 



S = .5^G 



(252) 



in which, when R and d are given in inches, (5 = 0.463. Cou- 
lomb gives the very inconvenient formula .S - — ^^ ■ 



Weisbach gives, from ver\- limited data, for wire rope : 
6 



S= 1.07S + 0.093 ~j^ 



(253) 



Exaiitple I. — Given a hemp rope 1" diameter, wilh a load of CGo lbs., bent 
■over a pulley 4" radius, from Eytelwein's formula we have : 

S80 

S = 0.463 = 101.8 lbs. 

4 
^'hich seems very high. Coulomb's formula gives 66 lbs. 

Ej:ajnple 2. — \ wire rope, composed of 36 wires, each 0.039" diameter, with 
Q. load of 550 lbs., is bent over .1 pulley 44 inches diameter. From Weisbach's 
formula we get : 

S = 1.078 -I- 0.093 = 3.403 lbs. 

22 

The utility of these formulas is doubtful, and a fuller investi- 
gation of the subject is much to be desired. It will be seen 
from formula (253) that for wire rope the value of R should be 
taken still greater than already considered for bending stresses 
(formula 246) ; this subject is also discussed in Chapter XXI. 

The above rules are deficient in that they do not take into 
account the kind of mechanical work absorbed by the stiffness 
of ropes. The angle embraced by the rope is, in the investiga- 
tions of Amontons, Navier, Poncelet and Morris, assumed to be 
constant, while in practice it is constantly changing, and exerts 
a. very material influence upon the result. 

The author's consideration of the subject is here given : 
Referring to Fig. S23, it will be seen that the fibres or wires 

on the concave side of 
the rope which passes 
over a pullej^ i?, are 
compressed, produc- 
ing a reduction in the 
form of the convex 
side, the compression 
originating with the 
load Q, being trans- 
mitted along the 
whole length of the 
twisted strands. The 
bent position of the 
rope can no longer 
retain its original sec- 
tion, of diameter d, 
but its volume must 
be the same as that 
Fig. 023. gf ^ corresponding 

length of the straight portion. The alteration in cross-section 




The details are as follows : 
Depth jr. Dia. ofwirea. Weight Go, S. 
3936 0.1043 ^-52 23,2'o 

32S0 0.09S4 1.36 22,gio 

2624 0.0925 1. 19 22,820 

1968 0.0S66 i.oc6 22,860 

1312 0.0807 0.912 23,130 

656 0.074S 0.785 23.690 

The twisting of the rope was commenced at the small end, and the diam- 
•eter of wires increased every 5 meters (16.4 ft) after ihe first 200 metres (656 
it). These ropes are very satisfactory, and last 3 to 4 years. 

* Conical drums are used in the American anthracite coal mines. 
+ See Dwelshauvers-Dery in Cuyper's Revue des Mines, 1S74 ; also F. Krane 
in Zeitschrift der Berg u. Hutteuwesen, 1864. 



s. 


s-)-i. 


27,260 


50,470 


25,710 


48,620 


24,170 


46,990 


22,640 


45,500 


21,090 


44,220 


19.550 


43,240 



may be of two kinds ; first ; uniform compression ; second, 
when this has reached its limit, a flattening of cross section. 

Both deformations are observed in practice. Ropes which are 
very flexible are loosely twisted, and therefore readily com- 
pressed as they pass over puUej'S. The general compression 
due to the tension r>f the load in the straight portion causes the 
twisted strams to pre -s firmly together towards the axis, so that 
a heavily loaded rope is very har<l. The compression is gener- 
ally permanent, and not elastic, as may be deduced from the 
permanent reduction in diameter of ropes after use, and is gen- 
erally due, in the case of wire ropes, to the compression of the 
hempen core ; as is shown by the observations of Leloutre and 
Zuber % 

The preceding remarks have not considered those wire ropes 
with metallic cores, used for running transmissions. Such ropes 
are always very stiff, and permit little or no compression. 
(According to Ziegler's experiments, only 0.22 to 1.2 per 
cent.) 

It is really almost as important, so far as flexibility is con- 
cerned, that a rope should have a suitable soft core as that it 
should be made of the best and most elastic and flexible mate- 
rial. This is shown by the fact that even with ropes made en- 
tirely of hemp or of cotton, and used for transmission over 
pulleys, the inner fibres, which are never in contact with the 
pulle3'S, show great wear. This wear is evidently due to the 
friction of the fibres against each other, due to the flattening 
and changes of cross section. For this reason the desirabilit}', 
or rather necessity, of lubricating the wires or fibres is evident, 
and this reduces the friction of the inner-lying portions of the 
rope. Rieten & Co. state that in the case of cotton ropes, "the 
rope always wears out by the internal friction of the strands 
upon each other, and that a load-twisted rope becomes useless 
in a shorter time than a soft, loosely twisted one, although the 
actual strength of the latter is the smaller." 

In view of all these conditions the insufficiency of the exist- 
ing rules for stiffness will be evident. It is apparent that the 
angle of contact must have a strong influence, and au entrance 
is found in cable roads where, when the cable is deflected 
through a small angle, small guide rollers are satisfactory, while 
much larger ones are necessary for greater angles. At a certain 
angle c, the deformation of the rope begins, and at another 
angle a maximum is reached, beyond which the resistance of 
stiffness is no longer dependent upon a. These points are of 
greatest importance with wire rope. It must be expected that 
the value of .S' will depend upon two functions of n, one for 
compression, and one for flattening. The first may be unim- 
portant with old and compressed ropes, the latter will be much 
dependent upon the lubrication and upon the coefficient of 
friction. 

I 269. 

Rope Connections and Buffers. 

The connection of one rope with another, when a smooth, 
junction is required, must be effected by splicing. This may be 
accomplished by the short or German splice ; or by the long, or 



?, 



M 



5 



G 



5' 



4' 3' 

Fig. S24. 



2! 



Spanish splice. The latter is the form to be used for wire rope. 
From the middle point i//of the splice, Fig. 824, if, for exam- 
ple, a six strand rope is in hand, strands i, 2 and 3 on the left 
are unwound, and strands 6', 5', 4', of the other rope wound in, 
and the ends cut and worked in. The same is done on the 
other side, the whole length i — 6 being 30 to 50 feet. 

To connect the end of a rope to another portion of the con- 
struction, the so-called hangers are used ; three forms being 
shown in Fig. 825. At a is tlie so-called "swan neck," which 
is secured to the rope by through rivets ; b is made with a coni- 
cal socket, the wires being douljled up, and soft metal melted 
and run in ; c is Kortum's hauger, the rope being held by two 
toothed w-edges driven in, and secured by pins. Numerous 
tests have shown this fastening to be as strong as the rope 
itself. 

In Fig. 826^ is shown a buffer coupliug used in the Zeutrum 
mine at Eschweiler, designed bj' the superintendent, Oster- 



X See Eeloutre, Transmissions et courroies, cordes et cables. Paris, Tignol, 
18S4. Ziegler, Erfahrnngs-resultant iiber Betrieb and Instandhaltung der 
Drahtseiltriebe, Wiuterthur, 1S71. 



lS2 



THE CONSTRUCTOR. 



kamp. The wrought iron thimble in the bight of the' rope 
:is fitted with a wooden block. Fig. 826 b shows the so-called 




Fig. 825. 



"friction hanger," both this and the previous form being 
arranged to be built into the upper part of the hoist cage. 




Fig. S26. 



In Osterkamp's design the spring cage is built into the yoke of 
■f-he frame, thus economizing room. 



I 270. 

St.^tionary Chains. 

Chains may be considered as jointed rods. Running chains 
are composed of very short members, in order that they may 
the easier pass over sheaves, while stationary chains, which are 
used in bridge, and other numerous constructions, arc made 
with quite long links. 




^■* 



Fig. 827. 



Fig. S27 shows the Admiraltj- form of stationary chain. Tha. 
links are made ]i fathom long, not including the thickness of 
metal, and are divided into 10 fathom lengths, each length con- 




sisting of 20 links. The lengths are joined hy a pin connec- 
tion, shown on the left, and the pin is made of steel, galvanized. 
Another form, known as the 
Gemorsch chain, is shown in 
Fig. S28, and is well known in 
Germany. 

Each long link is made 1.5 
metres long, and these are co 1- 
nected by short oval links. The 
coupling link is secured by a 
common, but heavy screw bolt. 
The proportions in the illus- 
trations are given in terms of 
the diameter of the rod. 

In order to enable such chains 
to hang freely, the so-called 
"swivel" is used. A heavy 
swivel, for chains such as Fig. 
S27, is chosen in Fig. 829. 
The swivel bolt has a ring 
attached which can be readily 
opened, and is large enough 
to receive two chain links, 
while the upper ring can re- 
ceive three. The limit of di- 
mensions is the thickness of Fig. S29. 
metal of the chain of Fig. S27. 

? 271. 

Running Chains. 





^=3;5^-^ 



'-'iii,~ 



The most important forms of running chains used in machine 
construction are those shown in Fig. 8jo ; a is an open link, and 
5 is a close link chain ; c is a stay link chain, and d a flat link 
chain. This latter is especially' suitable for a pitch chain, on 
account of the parallel pins which are at uniform distance from 
each other. The other three forms are made with a higher 
order of linkage, viz. : the globoid form already discussed in 
Fig. 224. 

In the wide open link chain a the globoid action can readily 



THE CONSTRUCTOR. 



183 



be disarranged ; less so in the close links of b, and hardly at all 
in the sta3--liuk chain r, which latter closely resembles the 
globoid link of Fig. a, p. 142. 

The proportional dimensions of chain links are not very 
closely determined. Those given in b and c are from the Ger- 
man Admiralty. The British Admiralty gives both for open 
and for stay-link chains, the pitch length 4 d, and width of link 
3.6 d ; in I'Vance, for open chains the length is made 3.25 </, and 
width 3. 4 d, and for stay-link chains, 3.S5 d and 3.75 d respec- 
tively.* 

In crane and hoisting machine construction, a vcrv important 
feature is the calibrating or adjusting of the linl.j of chain. f 
This is also a matter of much importance in connection with 
the chain propulsion of boats used in France and Germany. 
The chain used on the Sweetwater canal at Suez was made with 



d = 



16 



and a pitch of 3 (/, ;>nd breadth 3,2 d. The Magde- 



burg-Bodenbacher chain is very strono 



d being given - to 
16 



I's'', the links being proportioned as at b. 

Flat link chains have been used by Neustadt, made of multi- 
ple plates (see \ 94). The plates are made of the best quality 
and the pins made to project a little, and riveted over cold. 
Chains of this sort are also used for driving where heavy resis- 
tances are overcome, as in wire drawing, and in some fpinuiug 
machinery. 

I 272. 
C.-^LCUI^-iTIOXS FOR Ch.^INS. 

The chains which are made at the best establishments are 
always thoroughly tested, ever}- link being subjected to a stress 
closely within the limit of elasticity, or in some cases, slightly 
exceeding the elastic limit. A few links, usually three, are 
taken at frecjuent intervals every few weeks, and broken in the 
testing machine. The usual proof-load is such as to give the 
following stresses : 

5= 20,000 lbs. per sq. in. for open link chain. 
5'^ 25,000 " '• " " stay " 

The tests of chain for the German navy give for S: 

17,000 lbs. test of elasticit}-, 1 r ,■ , 

,„' „„ .. 1,- t J- < t r tor open hnks ; 

19,000 highest test, J ^ ' 

25,600 " proof load, | 

38,400 " breaking load for - for staj'ed links. 

three links, J 

For hoisting chain the elongation should be considered, and 
the metal should show an elongation before rupture of upwards 
of 20 per cent.j 

The permissible working stress per square inch section in 
Germany \ is ; 

9,000 lbs. 
I ^,000 lbs. 11 



For open link chain : 
For slay link chain 



From these we get for the proper total load P: 

For open links, P^ 14,000 d' \ 
For stay links, /"= 21,000 (/- ] 



(254) 



Flat link chains are subjected to the heaviest stress at the por- 
tion which is in engagement with the toothed chain-wheel. 
(See Fig. S37.) For this reason there should be not less than 
five link pins in gear with the Vv'heel at any time. If we assume 
that the tooth pressure is in arithmetic progression as i : 2 : 3 : 4 : 5 
the pressure on the body of the last pin will be \A, Pi and on 
each journal also ' ;' P, they being impelled forward by y'z P. 
If we put as a maximum stress in the bolts of 17,400 pounds,* 
we have for the thickness of plates <', pin diameter d, and num- 
ber of plate i, for a given load P, the following values ; 



* The length of fiat links in Fig. S30 a is given as 5 -{- 2.S a, and the projec- 
tion of the ends as 2 4- 1.4 d. These are in millimetres, and for inches the 
values o 1S75 + 2.3 rf, and o.oS -I- 1.4 rf should be used. 

I Excellent pitch chain is made at the Guttehoffi^ungshiitte at Oberhausen ; 
also by Scihlieper at Iserlohn, and by Dori^mieux at St. .\rnaud, and Plinchon 
Havez at Guerigny, and by Hawkes Crawshay at Gateshead on T3'ne. 

\ .-it the Guerigny Works the required elongation is : 

For rods I '2" to 1" iS per cent. 

For rods 1" to J.2" 16 per cent. 

For rods -;s" 14 per cent. 

For rods yV 12 per cent. 

For rods V-i" 10 per cent. 

I At the Gutenhoffnunghiitte. 

II Henry R. Towne, Treatise on Cranes, Stamford, Conn., gives a permissi- 
ble stress of g.ooo to 10,000 pounds. 



(S = 



0.0107 



^P 



/ -1- 2 ., — 
d = 0.0063 —— ^ P 



=. 0,5s (/ -f 2) 



(25s) 



The thickness R is made the nearest convenient value, and x 
must be a whole even number. For the latter we may take the 
nearest whole number to the value given by the relation : 



:0.26 



</ 



P 



(256) 



The following table has been calculated from these foruiuhe. 
The metal for the plates should be especially tough. Neustadt's 
chains had an ultimate resistance of four to five times the work- 
ing load. 

Example I. — An open link chain of i'' iron, according to (254) should have 
a working load P= 14,000 lbs., while a stay link chain of the same iron 
would permit a working load P = 21,000 lbs. 

Example 2. — Required to proportion a flat link chain to carry 22,000 pounds. 
"We have from (256I i = 0.26 -^Z P = 0.26 -^22,000 = 7.32, say S. Then in 1,255) 



0.0107 \/ 22000 



= 0.176" also (f = 0.0063 ^^ \/ 22000 = 

o -h I 



1.04". The pitch 



S+ I 
length / = 0.1875 \ 2.S X 1.04 = 3.1"; the width of plates = 2.6 X 1.04 = 2.7" ; 
the length of the body of the pins is 0.25 -f- 1.67 X 104 = i.gS, say 2", the 
diameter = 1.2 X i 04 = 1.25" and the length of link beyond the pin centre 
is o oS -t- (0.9 X ^.04) = I.02 say i". 




I 273- 
Weight of Chain. 
The length 5' of rod required to make a chain of a given 
length L bears the same relation to L as the length j for a sin- 
gle link does to the pitch I. We have for the chains a, b, c of 
Fig. S30 : 

Open Close Sta}' 



S 

"d 
s 

T 



2.52 



Links. 



9.42 



2,69 



Links. 



ri.94 



2.39 



Stay Links, 
including- stay. 

13-25 



2.65 



From these relations the weight of iron rods required may be 
determined (see \ 82). The greater the pitch of chain for a 
given weight of iron, the more economical is the form of con- 
struction. 1]' 

The load length and rupture length for chains (see \ 267) have 
been extended since that subject has been given practical con- 
sideration, this being especially the case with anchor chains (see 
next section). For this we may take the modulus of rupture 
A'at 37,000 lbs. for open links and 38,000 lbs. for stay links, 
with a modulus of safety T^ 20,00c and 24,000 lbs. respectively. 
We then have : 

T 

Li = and 

6.29 X > X_f_ 

/ 



6.29 X >' X 



y beiug the -weight of 



cubic inch of wrought irou =^ 0.27 lb., and heuce : 



1[ The pitch for stay link chain in the German navy ■ 
but has recently been made 4 d. 



;as fonnerU' 3 tf; 



1 84 



THE CONSTRUCTOR. 



Lt 



open Links. 
4672 
8672 



Close Links. 

4377 
S127 



Stay Links. 
545S 
8567 



Chains must also be provided with hooks for attachments to 
the load to be raised. 



Ch.\in Couplings. 

Chains which are used for transmission of motion (so called 
"endless " chains) require devices for coupling, as do al.=o those 
constructions with which chains are to be connected, and hence 
we have a variety of eyes, rings, coupling links, swivels, and 
the like. 




Fig. 832. 



A piece which is sometimes used with anchor chains is the 
so-called "twin" link, Fig. 831. This may be made of cast 
steel, and because of limited space is formed with circular open- 
'.ngs. The 9rdinary coupling link is shown in Fig. 832 a. The 
link isof wrought iron, the bolt and pin of steel, both galvanized. 
The pin is shorter than the diameter of the eye, and is secured 
on both sides by a plug of lead. The next link is made some- 
what longer than the other links of the chain, so that the 
coupling link may be more readily introduced. This form is 
used for joining pieces of chain to form greater lengths. The 
German Admiralty anchor chain is made with stay links, in 
seven lengths of 25 metres (82 feet) each, joined with coupling 
links, two of which are swivels. A bow anchor chain is given 
two more lengths of chain and made of iron 3inm. (o.iiS") 
thicker.* 

The chains for the system of boat propulsion are fitted with a 
coupling Mnk with rounded edges, and two are used together, 
as in Fig. S32 b^ which shows the chain u.sed on the Elbe. This 
coupling might also be suitable for power transmission chain. 

The swivel is used to permit the chain to have a rotation 
about its axis of length without twisting the links together. 




Fig. 834. 

A single hook is given in Fig. 834 «, and a double hook at 
Fig. 834 b. The construction of such hooks demands the great- 
est care, and according to Glynn, more lives have been lost and 
damage incurred by the breakage of hooks than by any other 
part of a crane. The case is one of combined resistance and 
leads to unexpectedly great dimensions. 

The diameter d^ of the shank of the hook may be obtained 
from formula (72), so that we have for a load P: 



(/j = 0.02 •■y P 



(257) 



This is based upon a stress of 3500 pounds, but an angular 
pull may increase this five-fold. Taking d^ as the unit, we may 
obtain the proportions given iu the illustrations in the following 
manner. Let lu be the width of the opening of the hook, and 
h the width of the body of the hook, the thickness at the same 
point is made -3 //, and for a stress of 12,800 lbs. upon the metal 
of the hook we have : 

— - = 1.30 \ -^-\-^^ or = 0.026 

o'l -^ /i ' 4 — 



VT 



V^ 



-+- 



(25S) 



The thickness at the point of the hook is made — , and hence 

the outside of the hook is a circle of diameter D = w -\- 1.5 h. 
We then have for : 



C.6 0.7 o.S o.g 




Fig. 833. 

The form of swivel used in the German Navy is shown iu Fig. 
833 a, and at Fig. 833 b is shown the English swivel. 




= 1.77 1.82 1.S6 1. 91 1.95 1.99 2.03 2.0S 2.12 2.16 



— :^2 = 0.035 0.036 0.037 0.038 0.039 0.040 0.041 0.042 0.042 0.043 
■^ P 



1.06 1.27 1.49 1.72 1.95 2.19 



2.44 2.70 2.97 3.24 
4 



= '3.72 4.00 4.28 4.59 4.88 5.18 5.48 5.S2 6.13 6.48 



The most useful ratio is 



I. In wharf cranes a weight 



is often combined with the hook in order to facilitate the 
lowering of the empty chain. This is shown in the dotted lines 
in Fig. 834^. In the case of a double hook each portion is cal- 
culated for its component j°, of the entire load P. From this a 
special unit d^' is obtained only for the dimensions w, /i, and D. 

Exaviplc. — Let the load upon a hook be .1400 lbs. We have from (257) /fi 
= 0.02 \/ /*= 0.02 \/ 4400 = 1.326". If v/e take lu = /: we get from the above 
h = 1.99 X I 326 = 2.63S'', and w is the same ; wliile D = 2.63S -f- 3 957 — tA", 

In the case of a double hook the angle between the components is 60° ; we 

^, , n 0.5 P 2200 

then have P\ — 



cos 30^ 
If we make -7- = 



0.866 
- o.g, h = 



= 2540, whence d\ = 
1.91 and 2v = 1.72. D 



>.02 \' 2540 = I.OoS 

= 1.92 + 2.86 = 4.5S. 



* The lengihs in the English Navy are 12;^ fathoms. 



For the upper portion we have as above <fi = 1.326". 



THE CONSTRUCTOR. 



185 



I 275- 

Chain Drums and Sheaves. 

Chain drums aud sheaves are usually made of a radius R = 
10 to \id, measured to the middle of the chain. lu some cases 
a rim is made ou the chain sheave, as in Fig. 835 a. 



whence we get, for 

.i = S 9 10 12 14 16 iS 20 

-^ = 1.3066 1. 4619 1.61S 1.932 2.247 2.563 2.879 3.106 




%m 



Fig. S35. 



This form of sheave brings a bending action upon the links as 
shown in Fig. 835 b. Sometimes the flanges are omitted and the 
edges of the sheaves bevelled as in the dotted lines, and in other 
cases the links have a bearing as shown at Fig. 835 c, in wliich 
the bending action is somewhat reduced. The bending is en- 
tirely avoided, however, by the use of a pocketed sheave, as in 
Fig. S36. 



The minimum number of teeth is 8. 
Neustadt uses the following : 

z ^= 8 for P = 500 to 6,000 pounds. 
.3" = 9 for /"= 6000 to 50,000 pounds. 
£■ =• 10 for /'= over 50,000 pounds. 

Guide sheaves for either kind of chain are made with 16 to 
30 teeth. 

For chain propelling cables ordinary smooth drums with 
parallel axes are used, with a groove for the chain. 

In Fig. S38 a is shown a section 



D I- - F 

^x'^'^^j^ \<ss\\"^\ xTc^'iv t.^\^^^^ s 




Fig. 836. 

This form is useful both for'chain transmission, and as a sub- 
stitute for winding drums in hoisting machinery, as it enables a 
small pocketed sheave to serve instead of a large drum. When 
such a sheave is made with only four pockets, they form a 

square with a side D' = I ■\- d -\- 2(1— d) v/o.5= 2.414 I— 
0.414 d\ while the side of the square of the alternate links is 
D" = 1.414 / -f- 0.414 d. The first gives the minimum, and the 
second the maximum, (double) lever arm with which the chain 
acts upon the sheave. If the pockets, instead of 4 and 4 are : 

6 and 6, we have D = 3.732 / — 0.264 d 
SandS, " " i? = 5.026 /— 0.198 a'. 

Chain sheaves of this form require accurately made pitch 
chain. 

When the load is heavy, the friction causes the chain to cling 
to the sheave, aud a stripper 5, Fig. 836, is required to lead the 
chain off in the proper direction F, while the entrajjce is pro- 
perly effected by a guide channel E. 

For flat link chain, a toothed chain wheel is used, Fig. 837. 




Fig. S37. 

In this form a guide channel ii, and stripper S, should also be 
used. The tooth profile is a circular arc with its centre at the 
link pin. If z, be the number of teeth, we have for the radius 
J', of the pitch circle : 

y= --^ ...(259) 





Fig. 83S. 



of the rim 01 the drum on the chain propelling gear on the 
river Elbe. This is made with steel flanges and channels on a 
wrought-iron rim. The last channel is made slightly larger it 
diameter in order to give a higher velocity to the driving sideo/ 
the chain. The wear upon the chain is an important item. Fig. 
838 b, shows a link of a chain as worn after long service. It 
must not be overlooked that the winding around the drum j,ro- 
duces a twist in the chain, givingasmany half twists in the chain 
as there are half convolutions about the drums. This twisting 
is not injurious if the chain is bent as frequently in one direc- 
tion as in the opposite. In fact, however, the chain is fjsually 
bent into more concave than convex bends. This causes a twist- 
ing motion to the chain and as it drags upon the bo'.tom and 
banks of the stream it produces much wear, and causes kinks to 
be produced at the shallow places. The chain mus', therefore 
frequently be raised at such points and a link opened and the> 
twist taken out. 
- This twisting may be 
prevented by using the 
drum arrangement 
shown in Fig. 839. This "" 
consists of simple 
drums all lying in one 
plane driven by gear- 
ing so that the proper 
relative motion is com- 
pelled. 

Fig. 539. 



? 276. 



Ratchet Tension Orc^ns. 



Tension organs may be combined with pawls, which in the 
case of cords are friction pawls, {\ 24S, 249), and for chains are 
toothed pawls, acting upon the links iti the same manner as 
upon ratchet wheels and ratchet racks. 

The establishment of Felten & GuiUeaume, at Miilheim a. 
Rhein, have devised a grip pawl for boat-cable driving, in which 
the rope is clamped to and released from a driving drum by an 
evolute shaped thumb clamp, the shock being reduced by a 
spring buff"er. 

Pawls for chains may be found used in connection with the 
heavy bow anchors of large vessels ; Bernier, of Paris, has also 
used such devices upon chain hoisting machinery. 




1 86 



THE CONSTRUCTOR. 



CHAPTER XX. 

BELTING. 

?276. 

Self-Guiding Belting. 

Belt p^ulleys are indirect acting friction wheels (? 191) and the 
belt itself is a tension organ combining the functions of driving 
and guiding (? 261J. Those belts which act without requiring, 
the use of special guiding devices may be called self-guiding 
belts. This action is attained by the use of cylindrical pulleys 
when the edge of the prismatic belt runs in a plane at right 
angles to the axis of the puUej- ; or in other words, when the 
middle line of the advancing side of the belt lies in the plane of 
the middle of its pulleys. 

When a belt runs upon a conical pulley in a direction normal 
to its axis, its tendency will be to describe a conical spiral path 
upon the pullej', as will readily be seen upon the examination 
of the development of the surface of the cone, Fig. 840. 



The leading off angle may be made as much as 25°, which 
occurs when the distance between the axes is equal to twice the 





Fig. S41. 



If the pulley is made with a double cone face or a rounded 
face, Fig. S41, the tendency will be for the belt to run at the 
middle of the face even when the direction of the belt is not . 
exactly correct. 

For leather belting, with a height of the crowning or curva- 
ture of the face s = J^ of the width of face, the belt may devi- 
ate from the plane of the pulley hy iY2° (tan = four per cent ), 
while for cotton belting, on account of the lesser elasticity of 
the material, the crowning s should not exceed -fij of the face, 
thus reducing very materially the permissible deviation. In 
ordinary circumstances at least one of a pair of pulleys should 
be made with rounded force. 





( E!..-h'i3 



Fig. S43. 




diameter of the largest pullej'. Another rule for the minimum 
distance between shafts for quarter-twist belts is to make the 

distance never less than ^ b'jD. 




Fig. S4?. 

The simplest arrangement of self guiding belting is that for 
parallel axes, Fig. S42 a and b, a being for open belt and b for 
crossed belt, eHher arrangement being suitable to run in either 
direction. 

For inclined and intersecting axes self-guiding belts are not 
suitable, except in the case of inclined axes in which the trace 
.S 5, Fig. S43, of the intersection of the planes of the two pullej-s 
passes through the points at wliich the belt leaves the pulleys. 
The leading line then falls in the middle plane of each pulley, 
but the following side of the belt does not, hence such systems 
can only be run in one direction. The leaving points'in the 
figures are at a and b^. The arrangement gives an open belt 
when the angle ft between the planes of the pulleys = 0°, and a 
crossed belt when /? = 183°. In the intermediate po.sitious a 
partial crossing of the belt is produced. If /J = 90°," the belt is 
half crossed (or as commonly called, quarter twist) ; if /i = 45°, 
it is quarter crossed.* 



i 277- 

' Guide Pulleys for Belting. 

When a belt transmission is arranged with guide pulle3-s, the 
proper guiding action is obtained when each guide pulley is 
placed at the point of departure of its plane with that of the 
next following puUey.f 





Fig. S44. 

In Fig. S44 examples are given of guide pulleys for parallel 
axes', all three pulleys l5"iug in the same plane. 

At a is shown a belt transmission with tightening pulley, b is 
a device for transmitting motion when great difference of speed 
is desired. In this case the guide pulley C\s as large as the 
driver A, and if desired may also be arranged to act as a tight- 
ener..!: At c is Weaver's device for similar uses.? In this case 
two belts are used, and the device has been used for driving 
circular saws. The pulleys should be fitted to run very smoothly 
in such devices. 

The cases in Fig. S45-846 have parallel axes with two guide 
pul]e3's. In the first case the guide pulleys are placed in planes 
tangent to both operating pulleys, and hence driving may occur 
in either direction. Usually, however, it is required to provide 



* The above geometrical construction is only approximate ; for an exact 
solution see a paper by Prof. J. B. Webb, Trans. Am. Soc. Mech. Eng'rs, 
Vol. IV., 1SS3, p. 165. ° ' 



t See also the paper of Prof. "Webb, referred to in the preceding note. 

i Eckert's patent (German) for driving the drum of a threshing machine 

g See Cooper's Use of Belting, Pliila., 1S7S, p. 171. 



THE CONSTRUCTOR. 



iS; 





^^^^.. 



Fig. S45. 



Fig. 846. 



for motion in but one direction, in which case the second form 

is used as being simjjler of installation. The pulley B may be 

used as one of the guide pulleys, 

in which case it may be placed 

loose upon the same shaft as A, 

and C or D be made drivers or 

driven. 

By placing the guide pulleys 
between the axes of ^ and B, 
instead of beyond them, they 
will revolve in the same direc- 
tion, and may be made fast 
upon one shaft, as in Fig. S47 ; 
this arrangement admitting of 
motion in only one direction. 

In Fig. 84S is an arrangement 
for inclined axes, which is a 
modification of Fig. S46, as will 
be seen by the dotted lines. 
The guide pulleys run in oppo- 
site directions, but ma)' con- 
veniently be placed upon the 
same shaft. 

In Fig. S49 is shown an arrange- p o 

ment of quarter-twist belts with ' '' 

guide pulleys. One side of the belt is placed in the intersec- 
tion S S o'i the planes of the xvvo pulleys. From any point e 





Fig. S48. 

on 5' 5, the tangents c a and e b are drawn, and in the plane of 
these the guide pulley C is placed. This arrangement permits 
of rotation in either direction. 

Another arrangement for the same purpose is shown in 
Fig, S50. The side of the belt leading off from A is inclined 
towards B, the other side passing over the guide pulley C, 
which is in the same plane as A and 5 S. This arrangement is 
well adapted for driving a number of vertical spindles from one 
horizontal shaft.* 




Fig. S40. 



Fig. S50. 



Fig. S51 shows the general case for inclined axes. Two 
points c and r, are chosen on the line of intersection 6' ,9 of the 
planes of the two pulleys, and the tangents c a, c b, c^ a^, c^ b^ 




drawn, and in the planes of these tangents the guide pulleys 
Cand Ci are placed. Under these conditions the rotation may 
be in either direction. The arrangement shown in Fig. 85 2 
occurs when the line 5 S passes through the middle of one of 
the puUej-s. 




Fig. S53. 

A simplification of the general case occurs when, as in Fiji. 
853, the guide pulleys fall upon one and the same geometrical 
axis which is parallel to the axes of both transmitting pulleys. 
In this case the only inclination of the belt is that given to it 
by the guide pulleys. The rotation can be in but one direction, 
viz. : that shown by the arrows ; if the reverse is desired, the 
guide pulleys must be placed as shown in the dotted lines. 
If the inclination of the shafts is too great the belt will be 
liable to drop off when the pulleys come to rest. The use of 
guide pulleys involves special hangers, a practical form for 
which is shown in Fig. 854.I 



* An example is Jacob's grinding mill with 40 sets of stones : see Uhland's 
Praks. Masch. Koustr., 1S68, p. 83, 1869, p. 242. 



t Patented in Germany by the Berlin-Anhaltischen Maschinenbau-Aktien- 
Gesellschaft. 



THE CONSTRUCTOR. 



The vertical axis is provided with an oil hole, and is fitted 
"by a ball and socket bearing to the bracket D. The flange ou 
the lower edge of the pnlley keeps the belt from falling off the 




Fig. 854. 



Fig. 855. 



pulley when at rest. The form in Fig. 855 was designed by the 
author for the arrangement of Fig. 848, both pulleys being loose 
upon the wrought iron shaft. 

If the position of the shafts can be so chosen that the line 
^ iT touches at least one of the pulleys, the very practical 
arrangement showu in Fig. S56 can be applied. If the distance 





Fig. 856. 



Fig. 857. 



^ Cis great in comparison with the width of bel':, the pulleys 
Cand C[ can be placed side by side instead of over each other, 
Fig. S57, in which case round face pulleys should be used. 




C2- 



Wh^. 



'SrSifiSSiiSUMB 



Fig. 85S. 



By the use of a fifth pulley the preceding arrangement may 
"be so modified that two pulleys, i?i and i?,, can be^driven from 
one driver, A. This is shown in Fig. 858 as applied in a spin- 
ning mill, in which the pulleys i?, and B.^a.re on different floors 
of the building, and are also provided with loose pulleys.* 




Fig. 859. 
lu the arrangement of Fig. S59 the pulley A drives two 



parallel shafts, one of which intersects its axis at right angles, 
t"he other passing beneath. 




Fig. 860. 

Another arrangement, devised by the author, is given in Fig. 
860. In this case the following side of the belt is passed over 
an idler pulley, Cj or C.,, and a second time arouud the driver 
(see also Fig. 795) by which the angle of contact a is doubled, 
and the modulus of friction f/« (^ 264) increased. This may be 
called a double-acting transmission. The cross section of belt 
may be made j% of a single acting transmission, so that in spite 
of the increase of length an economy of belting is obtained. 
One of the guide pulleys may also be used for a tightener. 
These devices will also be considered in connection with rope 
transmission (Chapter XXI.) to which they are especially appli- 
cable. 



? 278. 

Fast and Loose Pui,i,eys. 

Fast and loose, or tight and loose pulleys, as they are some- 
times called, are generally used in connection with another belt 
transmission in order to throw the latter in and out of action, 
the belt being guided by a belt shifter, which by the means of 
forks or finger-bars, enables the moving belt to be shifted. 
These shifting devices may properly be regarded as guide 
pulleys, and are sometimes fitted with rollers, as shown dotted 
in Fig. S61, at c and r^.f 




Fig. 861. 



Fig. S62. 



It is preferable to have the loose pulley upon the driven shaft, 
since the belt then can be shifted with a gradual spiral action 
by the shifter /^ Fig. 85i. It is best for the driving pulley to 
be made straight face, or if two fast pulleys are used side by 
side on the driving shaft, these should have very slightly 
rounded faces, if the belt is to be shifted promptly and readily, 
and for the same object the shifter should be placed as close to 
the driven pulleys as possible. The loose pulley should be kept 
thoroughlj' lubricated, and for this purpose numerous oiling 
devices have been made. The friction between the hub and 
shaft acts as a driving force upon the loose pulley, and this has 
been a Source of numerous accidents. This action is avoided 
in the arrangement in Fig. 862, in which the loose pulley is 
carried on a consecutive and stationary sleeve D.t 

A variety of mechanical belt shifting devices have been 
made, J the desire being to prevent the action of the belt 
from moving the shifter. A useful form is Zimmermann's 
Shilter, Fig. '863. 



•'^ See Fairba ., Mills and Millwork, II., London, 1863, p. 103. For the 
theoretical discussion of these various arrangements, see § 301. 



t Such rollers as especially necessary for shifting cotton belts, which are 
liable to catch on the shifter lingers, and even larger rollers are best in such 
cases. 

X See Berliner Verhandlun^en, iSfjg, p. 127. This has been used by the 
Society tor Prevention of Accidents, of JNIiilhonse. 

g See Berliner Verhandluugen, 1S68, p. 171, Rittershaus, Belt Shifters. 



THE CONSTRUCTOR. 



189 



The shifter bar F, to which the fork G can be clamped at any 
desired point, is operated by the lever H, which turns upon an 



and also a sin ji --= R — A*,, which gives : 




axis at/, forming a "dead" ratchet mechanism. The similarity to 
the ratchet devices of Figs. 754 and 755 will be observed. The 
movement of the bar is effected by connection at K or A'j. 




Fig. S64. 

Fig. 864 shows a shifter for quarter-twist belt. In this form, 
devised by the author, the guide pulley, which is required to 
support the belt, also serves as a shifter to move the belt to and 
from the belt pulley B, and loose pulley B^. If these pulleys 
are given greater width than that of the belt, as shown on the 
right, a vertical adjustment can be given to the upright shaft ; 
a condition sometimes required in grinding mills and similcr 
machines. 

I 279- 

Cone Poi,i,eys. 

When a number of pulleys are placed side by side in order to 
enable varied speeds to be obtained with belt transmission, and 
are united together in one member, we obtain what is called a 
cone pulley, such pulley being used in pairs. This construction 
involves the problem of determining the proper radii for the 
various pulleys, so that the same belt shall serve for all the 
changes, i. e., so that the length of the belt shall be the same 
for each pair of pulleys in the set. The problem may be solved 
as follows : 



!■#..' 





a. Crossed Belts, Fig. S65. The belt makes the angle ji with 
the centre line of the pulleys R and R^ ; and the half length of 

the belt, 1= R (-^+!^) + -^i (—+ P) + « cos /3, (! being 



the distance from centre to centre of the pulley. We then have : 



'={R + R,) (y + "^" ' 



(R+R,Y 



. . (260) 



This value is constant when R -]- R^ is constant ; that is, 
when the increase to the radius of one pulley is equal to the 
decrease in the radius of the other. Crossed belts are seldom 
used for this service, however, because of the injurious friction 
between the rubbing parts of the belt. 

d. Open Belts, Fig. 866. In this case we have : 

l=:(R + R^)^+{R-R^)fl + a cos ft 



Rr- 



{0 sin (3 + cos /3) + 



^sin/^l 



— {j3 sin (3 -f cos /3) sin /? I 

^ 2 . 



• (261) 



This function is transcendental, but may be graphically repre- 
sented in the following manner. Fig. S67. In the rectangle 
ABB' A', with a radius A B = a, strike the quadrant B M C 
about the centre A. Within this arc will fall all the values of 




/? which can occur. For any value oT (i = C A M, draw M N 
perpendicular to MA and make iI/iV = the arc M C =^ a ft 
Drop the perpendicular M P io A C, and draw A'' O perpendicu- 
lar to MP. NO will then = a p sin /3. Through A^ draw 
Q N K parallel to A B, and we have A Q = P O + A P=a 
(/3 sin [i -\- cos /?). By taking successively all the values of j3 
between 0° and 90° in this manner, we can determine the path 
of the point TV, which will be the evolute of a circle, C N D 

B D being equal to the length of the arc B M C ^ — a. If we 

2 

now draw D E parallel to B A, and take its middle point P, we 

a 



have Z? F ^ E F- 



and hence the proportion : 



£1 F : D B = — : — a = a : i^, and by similar triangles : 

T JiT^ — Q A = — (13 sin 13 + cos /3). 

This value is dependent upon — . If we prolong B F until it 
intersects A C prolonged, the resulting length A A' = B B' 
will bear to A' B' the ratio — . By then working B G = I, and 

drawing G H parallel to A' B', we have G PI ^ — . This 

TT 

I a 

length being transferred to /A' gives I T= — — — {j3 sin p 

TT TT 

-\- cos j3). We then have onl^- to use zh — sin /3 to solve the 

problem. 

Make A R ^ — , and we have the perpendicular R S= — 

sin ft By laj-ing this length off above and below T on G K, 
we obtain the points U and V^ and this finally gives / U for the 
radius R of the larger cone pulley and I V ^= Rj, the radius of 
the corresponding smaller cone pulley. 

By solutions for successive values of /?, we obtain the curve 
DUX T E, which can be used for the determination of the 
radii of any desired pair of pulleys, each pair of ordiuates 
measured from Pf I belonging to corresponding pulley on each 
cone. 

In practice it is usual to find one of the cone pulleys given 
and the dimensions of the other required. In this case V U 
may be taken as the difference R — A*,, between the radii, were 
the steps uniform. By taking this difference A" — A", in the 
dividers, and finding the equivalent ordinate U V on the curve, 
and then adding J' / = R^, the axis U I is found. 

In order to use the curve conveniently, it may also be laid off 
left-handed, as shown in the dotted lines D' X E'. 

The use of the diagram will be rendered still more convenient 
if we omit the unnecessary value /. This enables us to distort 
the curve in the direction of the abscissas to any desired extent. 



iqo 



THE CONSTRUCTOR. 




off toward C, the corresponding radius Xd and 
prolong the axial line d d' to its intersection d' with 
B E. Then Iiy off the given geometric ratio on C X, 
considering X d as i (shown in the diagram by the 
small circles for the ratios \, \, J, J, f), and draw rays 
from d' through the points of division, and these rays 
will intersect the curve at the correspouding points 
for the pulley radii Ry We then have for the radii : 

a I and a \' for the ratio i : 4 
b 2 " b 2' " "2:4 

c ?, " c z' " "3:4 

dX" dX" " 4:4 

e s " c i' " "5:4 

e " c 0' " "6:4 

Cone pullej s may also be made continuous, thus 
becoming conoids upon which the belt can be shifted 
to any point by au adjustable guide or shifter. Such 
conoids are used for driving the rollers in spinning 
machinery. Such a pair of conoids are shown in 
Fig. 869, the proportions having been determined by 
the graphical scale. The angular velocity varies in 
au arithmetical ratio as shown. 

The curve E i' A in the scale shows the limit to 
which the axial line may approach A E ; this dis- 
tance must not be less than R -\- R^=- a, from which 
V Y=\{AB— V U). 

? 2S0. 
Cross Section and Capacity of Belts. 

A belt of rectangular cross section of width b, and 
thickness (5, will be subjected to a tension 7" on the 
tight side (see | 264), which it must be proportioned 
to sustain. If^'isthe permissible stress for the unit 
of cross section, we have, therefore. T ^ b <! S. 

The minimum ratio which T bears to the trans- 
mitted force P is dependent upon the stress modulus 



But r = 



in which 




Fig. 868. 

This has been done in the proportional diagram for cone 
pulleys, Fig. 86S. 
The method of using the diagram is as follows : 
The sides A B and D E oi the rectangle represent the dif- 
tance a between the centres of the pulleys ; all radii are given 
in proportional parts oi a, for which reason A B is sub-divided, 
the size of the diagram being selected so that A B = 1 8 to 20 

inches. If, then, i a and 1' 
a are two given radii for a 
pair of pulleys on a pair of 
cones, we take the vertical 
chord of the curve which 
= 1' a — I a, prolong the 
chord downward until its 
length = I a, and draw the 
axis abed parallel to A E. 
Then for the other pairs of 
pulleys on the cones, we 
\ 2:3 t;!2 have bz and b 2' , c ■i, and 

cy, etc., which can be 
taken directly from the dia- 
gram with the dividers. If 
the given pair of radii to 
which the cones are to be 
made equal, the chord R — 
Ri =; o, and the axis will 
pass through X at right 
angles to C X. 

If it is desired to construct 
a pair of cone pulleys to any 
given speed ratio, this can 
readily be done. If, for example, the given ratio is i : i, we lay 



r, since T^r P('i 264). 

p—i' 

represents the modulus of frictipn c\/<'. Sence, if X 
is the horse power transmitted for a belt speed of v 

Pv bSSv 

feet per minute, we have : X = =r . 

33°°° 33000 r 

This enables us to determine the cross section of the 
belt, but in practice the width of the belt is the varia- 
ble factor, the thickness usually being determined by 
commercial considerations, and limited to few defi- 
nite sizes. 

If we let q represent the cross section of the belt 
in square inches, we have : 

g V S 
13000 T 

This formula is very useful, since it may be used to determine 
the capacit}' of a belt from its cross section and velocity. If we 

N 
put N^ = — we have : 



7V=- 



^o = - 



33000 



(262) 



The value depends upon the material and stress modulus, the 
latter including the arc of contact a, and upon f, which itself 
depends upon the material of both belt and pulley ; it may also 
be considered as dependent upon n, independent of the material, 
in the same manner as v.'as the subject of specific weight. The 
author has called this value vV^, the specific capacity o" a belt. 
It will be seen that when this specific capacity is determined 
for any kind of belt, the proper cross section for the transmis- 
sion of a given horse power X can readily be found, since the 
velocity v can be chosen, and we have at once 



_X_ 
X^v- 



(263) 



For the determination of the specific capacity of any kind of 
belt it is necessary to find the constants S and t. 
The materials used for belting are: 

Tanned leather. 

Cotton, woven and treated with oil, 

Rubber, interlaid with linen or cotton webbing. 

In practice the value of 6" to be used must depend much upon 



THE CONSTRUCTOR. 



191 



judgment, the value being governed to a great extent by the 
quality of the material. Customary values are for: 

Leather 5 =: 4000 to 6000 lbs. 

Cotton S^ 3000 to 4000 lbs. 

Rubber 5 = 3500 to 5000 lbs. 

The thickness i for single leather belts varies from j'5" to -<],/' ; 
double, triple, quadruple, and even quintuple thicknesses being 
sometimes used, the thicknesses being secured by cement, and 
sewed or rivetted togetlier. Cotton belts are usually from ■^■/' 
to W" thick, while rubber belts are made of any desired thick- 
ness, a web of can\'as being interlaid between the successive 
thicknesses of rubber. 

The stress modulus - depends upon a andy^ and the latter co- 
efficient varies with the age of the belt, being greater with belts 
which have been used some time than with quite new belts. It 
is advisable, however, to make all calculations as for new belts, 
in which case we have for smooth iron pulleys, for: 

Leather and cotton, y= 0.16 to 0.25*, p = 1.6 to 2.1 
Rubber, y= 0.20 to 0.25, /) = i.S to 2.1 

These give as approximate values for . 

T 
Leather and cotton, -— or r = 2.5 to 1.9 



"=■75 y- 3°° 
80 



Rubber, 



P 



By using these values together with those given for 5, in (262) 
■we get for the specific capacity for belting : 

Leather, iV„ = o.oo62"to 0.009S ") 
Cotton, N^ = 0.0036 to o 006S K (265) 
Rubber, N^ = 0.0050 to 0.00S2 J 

These are based upon low and moderate speeds ; say up to 
3000 feet per minute, and the variations between the Itmits given 
are those due to the differences in strength of various kinds of 
leather and canvas used. 

The resistance to bending or stiffness of a belt must be taken 
into account, and the ratio of thickness S to pulley radius 7?, 
must not be too great. Practical experience has shown that 

& I 

-=- = — should not be exceeded to obtain best results.* 
R 50 

From the known stress and the thickness of the belt the 
superficial pressure p, between belt and pulley may be cal- 
culated. We have only to substitute in (24r) for the width b' 
•of 'he surface of contact, the width b of the belt itself, and 
sin 1^ g = b (5, we get the simple relation : 



L 
s 



& 
H 



(264) 



Example /.—Required a leather belt to transmit 
of pulleys to be n = So «i = 150 revolutions. Taking 

0.007 ^"<i the lineal velocity of belt at 3000 feet, Tve have g 

sq. in. cross section. 

If we ust JL double belt 0.4" thick, the width should be - 



3o H. P. and the speeds 
the specific capacity at 

IGO „ 



3000 X 0.007 



For the drivins 



2 TT K 71 

■ pulley we have: — — 



e,andA'= 3°°°Xi2 ^ ^_ 



say 72", or 154 inches. Forthe driven pulley we have J^i = 



2 TT ^^ 
So X /'"s 



For the superficial pressure p, we have /^ — 
2750 



33000 X TOO 
3COO 



= 1 100 lbs. Also 



2:5 P= 2750, hence S^ 



= 573- We have also i = i s P - 1650, 



■ 0.4 X 72 
which gives Sn = 343, or a mean of 458 lbs., which in (264^ gives a mean value 

p = ^^—^'-— = 2.5 lbs. on the large pulley, and/ = ^^—^^-^'-^ = 4.9S, or 

72 33.4 

nearly 5 pounds. This is verified since, 'n f =' 0.16: 

72 X 3-^4 X 12 X 2.5 X o-*6 = iioo, 

-which is the value of Pas above. 

Fxample 2. — Whathorse power can be transmitted by a cotton belt 4 inches 
-vv:de and 0.25" thick, at a velocity of 2000 feet per minute ? Taking the speci- 
fic capacity at o.ooo, which has been found satisfactory in practice, we have 

from (262} j:V= q v I\'q = 4 X 0,25 X 2000 X 0.006 — 12 H. P. 

E-rainp/e 3. — A rubber belt is required to drive a centrifugal pump {rubber 
being especially adapted for damp locations). ,A''=2o, the pump vane to 
make 300 revolutions, and the driving shaft So revolutions per minute, and 
the belt speed 2000 feet. Taking the specific capacity at 0.007, we have 20 == 
q X 2000 X 0.007, hence ^ = i 43 sq. in., and if we make 5 = 0.2 we have a 

Tvidth b = 7.1". For the driven pulley we have Tvi = = 12. "i, say 

■z IT 300 ■' 



'• For cotton the thinner belts from 0.25" to 04" are preferable. 



= 47.S''. A mean value of 5 is 425 
6 7 on 



1.7S on the large pulley and -^— — 

12 71 



125^", and for the driver R = 

^u ^ J. ■'^3 X 0.2 
lbs., whence/ = ^i — 

47.8 

the small pulley. 

For extraordinary cases the fundamental formtila should 
always be applied. Foi double-acting belts, as in Fig. S60, in 
which a = 2 TT instead of ~, the valuey"a =^ i, and the modulus 
of stress is only o 6 of the preceding value, hence ^ is re(ltice<l 
in the same proportion. If the belt velocity 7/ is very high, it 
is no longer permissible to neglect the influence of centriftigal 
force. For a speed v = 5000 feet and a stress S ^= 5^^ pouuds 
(see ■§ 264) the exponent in the friction modulus becomes o.S4y a 
instead of _/" a, which iox f = 0.16 and a =^ tt, givesy"'' a = 0-S4 
X o.r6 TT ^= 0.42. This gives r ^ 2.91 or about -J of the normal 
value, which requires one-sixth greater cross section g for the 
belt. The highest limit of belt speed in ordinary practice 
appears to be about 6000 feet per minute.f 

? 2Sl. 

Examples of Belt Transmission. 

The table of existing examples of belt transmission on next 
page will serve to furnish data tor comparison with calculated 
results. 

The great variations in the values of .9 and N^, in the fol- 
lowing table are not surprising when the differences in the 
quality of material, and the various conditions are considered. 
Many leather belts are working under high stresses which are 
only practicable because of the excellence of the material. 
Some such belting can be operated under stresses as high as 
200O pounds, which enables much lighter sections to be used. 
Many belts which appear to have been excessively heavy have 
simply been calculated to work at a moderate stress. 

The plausible but erroneous idea that the pressure' of the 
atmosphere influences belt action cannot be admitted. It is 
contradicted not only by the fact that the same coefficient of 
friction exists for ropes as for belts, but also by the recent 
and careful experiment made in a vacuum by Leloutre 'vhich 
confirmed the theory of the modulus of friction. 

? 2S2. 

Belt Connections. 

The various methods of connecting the ends of belts generallj' 
give a greater stress at the point of connection than in the body 
of the belt. Tlie attempts to reduce this weakness and also 
provide for the greatest facility in the making of the joint, has 
caused a great variety of methods to be proposed ; some of the 
best of these are here given : 

a. h. c. li. 



ODD 

D 



TTTTJ: 

(J ») i 





riG. 870. 

In Fig. S70, « is a lap joint sewed with hempen thread ; b, a 
lap joint secured with screw rivets ; c is a plate coupling, the 
plate and prongs being made in one malleable casting and the 
prongs bent over and clinched after insertion in the belt, several 
clamps being used for belts more than 4 inches in width. At d 
is shown belt lacings for use with single or double belts. The 
upper one has the defect of giving intersections which make 
the lacing cut itself, and the knot at the edge of the belt reduces 
the strength of the joint. i These defects are both avoided in the 
lower form, which is an American belt lacing.^ 



flu the construction of tlie Anherg- tunnel a hoi.'^ting machine was used 
in which the belt had a velocity of 4700 feet per minute, which worked well 
for fourteen months. 

t Leloutre has used the fc-pper form of lacing for a belt of 26" wide, 0.66" 
thick with excellent performance and durability. 

f See Cooper, LTse of Belting, p. 1S9. 



192 



THE CONSTRUCTOR. 



1 










exajviples of 


BELT 


TRAJfSMISSIOK, 


t 




No. 


Horse 
Power N 


n 


R 


V 


p 


b 


<S 


S 


■ ^0 


REMARKS. 


1 


624 


40 
100 


271.8 


2887 


7II4 


105 


0.67 


512 


.0062 


Leather, 2 belts side 
side. 


by 


2 


200 


52 

1 82 


39-37 


3749 


I73I 


24 


0.47 


388 


.0046 


Leather. 




3 


190 


J5_ 
223 


71.S 
22.6 


2440 


2528 


•' 21 


0.24 


1222 


.0147 


Leather. 




4 


17s 


120 

T2'8" 


_56^ 
30 


3561 


1573 


29 


°-35 


388 


.0046 


Leather. 

1 


5 


153 


120 


J3_ 
47-25 


3955 


1256 


12.6 


0.52 


4S3 


.0059 


Leather. 




160 


6 


130 


36 

94 


1 28 

45-3 


2410 


1544 


10 


0.40 


981 


.0121 


Leather. 


7 


90 


65 
182 


83.8 
30 


2833 


IG34 


12 


0-35 


6l2 


-0075 


Leather. 


8 


Si 


55 
137-S 


98.4 
39-37 


2833 


928 


9-8 


0.52 


455 


.0056 


Leather. 
* 


9 


60 


100 


59 
59 


153s 


631 


■ 
12.2 


0.47 


270 


-0033 


Leather. 


100 


10 


54 


45 
125 


9&A_ 
35-4 


2318 


660 


17-3 


0.24 


400 


.0092 


Leather. 


11 


42 


60 
90 


70.S 
47-25 


2224 


614 


ii.8 


0.20 


. 654 


.00S2 


Leather. 




12 


40 


66 
102 


49-2 
38.9 


2066 


630 


13.8 


0.24 


4S3 


.0059 


Leather. 


13 


530 


60 
262 


144 
27 


5156 


3337 


-.^ 


0.72 


313 


.0036 


Cotton. 


14 


497 


144.4 


99 
48 


3620 


4457 


30 


0.72 


526 


.0065 


Cotton. 


15 


470 


62.5 
114 


96 

49-5 


3130 


4S77 


32 


0.72 


540 


.0065 


Cotton. 


16 


413 


120 


120 


3000 


4453 


30 


0.72 


512 


.0105 


Cotton. 


17 


325 


125 
172.4 


60 


3915 


2583 


22 


0.72 


412 


.0049 


Cotton. 


18 


134 


133-3 


4! 
45 


Z^Z° 


1390 


10 


0.72 


412 


.0049 


Cotton. 


19 


60 


70 
175 


_73£ 
29.6 


2706 


722 


16.S 


0,47 


228 


.0049 


Cotton. 


20 


35 


81.3 
99-3 


38 
31 


1633 


704 


5 


0.72 


498 


.0062 


Cotton. 


21 


66 


165 
243 


55 
37-5 


4763 


451 


1 1.8 


0.52 


185 


.0023 


8-ply Rubber. 




1 ■ 



THE CONSTRUCTOR. 



193 



Fig. S71 a, shows Better's belt fastening. This is a form of 
belt hook which has been found very serviceable, reducing the 
strength of the belt but little, and permitting easy renewal. 
Another form is Moxon's belt fastening,* shown at b, is a pin 



iMiymi 



-mHrtKHJt 




Fig. S71. 

point, the ends of the pin being riveted over, and from its con- 
struction should be very strong. At ^ is a butt joint with a 
reinforcement piece especially suited for cotton belts. When a 
belt is made for special service it can be in several layers as at 
d ; the joints overlapping, but thus giving no opportunity lor 
change of length. 

The stretching and joining of heavy belts is a matter requir- 
ing much care in order to secure the desired tension, = yi 
(7^-|- t)-\ Belts which are subjected only to light tensions may 
be cemented bj' scarfing the ends and using a cement composed 
of common glue mixed with fish glue, or of rubber dissolved in 
bisulphide of carbon, 

I 283. 

The Proportions op Pui<i,eys. 

Pulleys are usuall}' made of cast iron and of siugle width, /. e., 
one set of arms. The arms, which formerly were made curved, 
in order to resist the stresses due to contraction, are now made 
straight, and for wide face pulleys two or even three parallel 
sets of arms are used. 




- -| - 
i 

i 



Fig. S72. 

Fig. S72 shows both single and double arms. The dimensions 
of arms and rim have been determined by experience, based 
upon practical considerations. For the number A of arms for 
a single set, we get serviceable values from : 



which gives, for : 
R 



A= I'i 



3 4 5 



+ 



f) 



(266) 



9 10 IT 12 13 



^=34567 S 9. 

The width h of the arm, if prolonged to the middle of the 
hub, may be obtained from : 



h = 0.25' 



^ ^ ^ 10 A 



(267) 



The width It-^ of the arm at the rim is equal to o.S /;, and the 
corresponding thicknesses are e = % /i, and f, = jA A,. 

Pulleys with two or three sets of arms may be considered as 



* See Chronique industrielle, 1882, Vol. 5, p. 97 : also Mechanical World, 
1882, Vol. 12, p. 56. 

tLeloutre has used a dvnamometric belt-stretcher for tensions of Y- 
(r+ = 8800 pounds. 



two or three separate puUej'S combined in one, except that the 
proportions of the arms should be o.S or 0.7 times that of single 

f.'3"and ^yf 



arm pulleys, or in the proportion of 



The thickness of the rim may be made : ,i = i to ;4^ //, 
being frequently turned much thinner. The width of 



this 
face 



The thickness of metal in the hub may be made /F= /i, to 
;4 h. The length of hub may = b, for single arm pulleys and 
2 b for double arm pulleys. Light pulleys are usually secured 
to the shaft by irieans of set screws, as in Fig. S75 and 877 ; 
heavier ones are keyed as in Fig. 191, either with or without 
set screws.* 

For many purposes puUe^-s are made in two parts, such being 
commonly called "split pulleys. The forms of split pulleys 

are shown in Figs. 873 to S75. 
The arrangement of the two 
halves is clearly shown, that 
of Fig. 874 with hollow clamp- 
ing section, being especially 
good.t 

The form in Fig. S75 is the 
design of the Walker Mfg. Co. 
of Cleveland, Ohio, the clamps 
being made of malleable iron 
or steel. In all three cases 
there is no especial method of 
fastening to the shaft. In 
England and America pulleys 
are frequently made with 
wrought iron rims and cast 
iron hubs. This construction 
greatly simplifies the casting 
of the arms, and at the same 
time gives pulleys 25 to 60 per cent, lighter than those of cast 
iron, which in large transmissions greatly reduces the friction 




Fig. S73. 




Fig. S74. 

at the bearings of the shafting. Fig. S76 shows the Medart 

pulley. The rim is curved in bending rolls, and also given a 

rounding face, and is 
countersunk for the 
rivets at the attach- 
ment of the arms. The 
pads on the arms are 
truly finished, as is 
also the rim after it is 
riveted on, thus giv- 
ing an accurate and 
useful pulley .J 

A metal pulley by 
the Hartford Engin- 
eering Company 60" 
diameter and 16'' face 
weighed 320 pounds. 
A cast iron pulley of 
the same dimensions 
made by the Berlin- An - 
p bait Works, weighed 

'^' 700 pounds, and one 

by Briegleb, Hansen & Co., a little narrower face weighed 528 

pounds. 




^^ 



* In order to determine the necessary friction to secure a pulley to the 
shaft, the force ;^ on the belt will serve. In ordinary cases, assuming- a co- 
efficient of friction on the key of one-half that on the belt, there should be 
a pressure;^' on the key of about 4coo times that on tlie belt, which, accord- 
ing to ^ 20 will not give more than 5000 to 7000 lbs. for/'. 

fXhis is the construction of the Berlin-Anhalt Machine Works. 

X Made in England by George Richards & Co., Manchester. 



194 



THE CONSTRUCTOR. 





Fig. 877 shows Good- 
win's split pulley, with 
wrought rim, the face of 
the rim being rounded by 
turning. 

These constructions 
naturally led to the use of 
wrought iron arms also, 
although these are some- 
what difScult to make ; 
but for very large diame- 
ters (say 16 to 25 feet) 
they possess advantages.* 
Pulleys made entirely of 
steel are used by J. B. 
Sturtevant of Boston, in 
connection with fan blow- 
FlG. S76. ers, Fig. S7S. The hub 

"with web, is screwed on the steel shaft of the fan wheel, and 
the rim, which has a groove turned in it, is expanded by warm- 
ing, and shrinks into place, 
the whole being finally 
turned iu position, and care- 
fully balanced. Sturtevant 
uses these pulleys up to 10 
in. iu diameter, and 7 in. 
face, the thickness of rim 
being from 0.08 to 0.16, and 
the velocity at the rim reach- 
ing 5000 feet per minute. 

By covering the rim with 
leather the co-efficient of fric- 
tion, f, andean be increased 
between the belt and pulley, 
and the modulus of stress 
T reduced, and the specific 
capacity of the belt increased. 
This is sometimes useful be- 
cause a reduced mo. lulus of p o 
stress T permits a smaller ''' 
cross section of belt and lighter pulley. In large transmissions 
reduction of stress is important since it is accompanied with 

reduced journal friction 
and higher efficiency. 
The observation of the 
author leads him to be- 
lieve the specific capa- 
city of a belt is not 
greater with leather 
covered pulleys than 
with uncovered ones, 
and the cost of covering 
is an important item. 

The greater the angu- 
lar velocity of a pulley 
the more important it is 
that its geometric axis 
should be a so-called 
"free axis." This re- 
quires that the center of 
Fig. 878. gravity of the pulley 

should be on the axis of rotation and also that the various por- 
tions of the mass should be so distributed that the axis of 
inertia should coincide v;ith the axis of rotation and the centri- 
fugal moment equal zero f This can be done empirically by so- 
called balancing, the unequal distribution of material being 
■equalized by attaching pieces of lead or other metal, or more 
accurately by balancing when revolving, for which purpose a 
beautiful apparatus has been made by the Defiance Machine 
Works, Deiiance, Ohio. Careful balancing of pulleys is of great 
importance at high speeds, the rapidly increasing vibrations 
^ill soon limit the speed. This i= to be considered in connection 
with the advantages to be gained by the use of high speed 
shaft as discussed in ? 146. 

Note. — The recent investigations upon paper rim pulleys % 
are instructive. This construction gives a very high modulus 
of friction, the modulus of stress r being only 1.2. This gives 
T ^ 1.2 P as against 2.5 P, for iron pulleys. Hence follows a 
great increase in the specific capacity of the belt, and increased 
efficiency with smaller and lighter pulle3'S. This leads the way 
to further investigations which prove of material value in the 
science of belt transmission. 



? 2g 



r^s: 


♦\-S\\\>v\;;-S:^\ .sV '■**«¥l^'^-i\':.';^^ 


^N-\-s^\S*Ki»Nl^l 


^<^' 


^■.///^i'A'.y 


/■v//M/^////i,//////^/,'/'./y 






ri 


\\ 


\ 


i 


// 






* Pulleys with wrought iron arms are made in Germany bv Starck & Co., 
Mainz ; in Eng^land by Hudswell, Clark & Co., Leeds, these latter with 
arms of round bar iron. 

tSee an article by the writer, " Ueber das Zentrifugal-Moment," in Ber- 
liner Verhandlung, 1S76. p. 50. 

X See Am. Machinist, May 23, 1885, p. 7, 



Efficiency of Bei^ting. 



Three causes of loss exist in belt transmissions, viz. : journal 
friction, belt stiffness, and belt creeping. For horizontal belt- 
ing we have, according to formula (99) for the journal friction, 
expressed at the circumference of the pulley a loss E^ when 7 
=2.5 P, t = 1.5 P: 



^' -F - 



+ 
P 



t 4 



At^- 



^1^ - 



-^yi^i 



:). 



(26S) 



in which rfandrfj are the journal diameters, and/" the coefficient 
of journal friction. This loss is doubtless the greatest of the 
three. For lack of better researches the loss of belt stiffness 
may be deduced from Eytelwein's formula for ropes. For the 
coefficient of stiffness s, for force S', which includes both pul- 
leys ; 



-^ = E' =s 



T-\- t 



P 



V7? 



+ 



R ' y?i 



-.\s 



fi2 &"• 



) 



(269) 



in which i =: 0.009 — = 0.012. 



The loss from creep is due to the fact that the greater stress 
on the driving pulley over that on the driven requires for a 
given volume of belt, a longer arc of contact ; for the expendi- 
ture of force G' for creep on both pulleys, we have for a stress 
5i on the leading side of the belt : 



?--■ = 



J 



! + ■ 



E 



E^S^ 



(270) 



S, 



In this E is the modulus of elasticit}' of the belt, which for 
leather is 20,000 to 30,000 pounds. The losses from stiffness 
and creep are .small. 

Example. — l^et d and dx ^ a" \ R = R = 20", S = 0.2, _/ = 0.08, 5= 0.012^ 



ii = 28,440, Si = 425, we have F^ = P 



; X o.oS 



■ X 0.4 — 0.08 P; 



also S- = P (0.048 X 2) ■ = 0.004S /; 
20 

and Gi = /> "-' ^ ^-^ ■ = 0.0059 ^• 
28,440 -t- 425 

The total loss is therefore : o.oS -f- 0.0048 -j- 0.0059 = 9.1 per cent 

CHAPTER XXI. 

ROP£ TRANSMISSION. 

§ 285. 

Various Kinds of Rope Tr.4.nsmission. 

If in the tension driving gear, shown in Fig. Sio, the rope be 
used only for the transmission of power we have what is called 
a Rope Transmission. Since the details of construction must 
vary, according as fibrous or wire rope is used, we may distin- 
guish between three kinds of rope transmission, viz. : those for 
Hemp, Cotton or Wire Rope, and these will be considered in 
this order. The oldest of all these is hemp rope transmission, 
but this was gradually being superseded by belting until 
Combes, of Belfast, revived it, about 1S60, since which time it 
has been extensively used for heavy transmissions. The char- 
acter of the material permits a wide variety of applications. 
The same is true of cotton rope, which is extensively used for 
driving spinning frames, travelling cranes and many other ma- 
chines, the softness and flexibility of the material giving it ad- 
vantages, but within limits. Wire rope transmissions, since its 
introduction by the brothers Hirn, at Eogelbach, in 1S50, have 
developed a high degree of efficiency and utility for long dis- 
tance transmission. As will be seen hereafter, the applications 
of rope transmission appear to be capable of still further ex- 
tension. 



THE CONSTRUCTOR. 



I9S 



A. HEMP ROFE TRANSMISSION. 
I 2S6. 

Specific Capacity. Cross Section of Rope. 

It is importaut first to determine the specific capacity for 
ieinp rope (§ 2S0). This is obtained from the general state- 
ment according to (262) : 

N« = A ^, 
3 ^ 

in which 5'i is the stress on the tight side of the rope, and r the 
modulus of stress. The value for the co-eiScient of friction /, 
depends upon the form of the groove or channel in the sheave 
over which the rope runs. 




Fig. S79. 

If the groove is semicircular, as at b, Fig. S70, the friction is 
but little greater than it is upon an ordinary cylindrical pulley, 
as at £7 / if, however, the groove is made wedge-shaped, as at c 
(see wedge friction wheels | 196), the driving power is increased 
although the surface of contact is reduced. In determining the 
value of r, from formula {239) the iuflueuce of the shape of the 
groove can be included by using a correspoudiug co-efficient of 
friction /K According to the recent investigations of Leloutre 
and others, the value of^^for cylindrical pulleys and new hemp 
rope is 0.075, 'o'^ semicircular grooves, o.oSS, and for wedge 
grooves with an angle of 60°,/'= 0.15, which accords well with 
the action of the wedge, doubling the pressure, see (1S5). For 
y = o.oSS and a contact of a half circumference, we have y a 
= 0.3, and hence r = 3.S6 ; with y = 0.15, y ■ a = 0.47, and r 
^ 2.67. The latter value, which is even' reduced in actual 
practice, may be adopted, since wedge grooves in general use. 
The stress is usually taken while low, and may be put at 5 ^ 

I 350 

350 lbs., which, taking t = 2.67, gives N„ = . — - — = 

0.0039 ; see (262). In practice No is found even one-half this 
value, and we may take as a practical rule in hemp rope trans- 
mission for the specific capacit}', ;. e, the horse power trans- 
mitted per square inch of cross section, for e^ich foot of linear 
velocity per minute ; 



No 



■ 0.004 to 0.002 (271) 



the cross section being taken as in ^ 265, as that due to the full 
outside diameter of the rope. 

When great power is to be transmitted a number of ropes are 
used side by side, the pulleys being made with a corresponding 
number of grooves. For machine shop transmission such 
Topes are conveniently made about two inches in diameter, 



although they are 
inches. 



used as small as i}{, and as thick as 2i{ 



Example i. A steam engine of 60 H. P. has its power transmitted through 
"five ropes of 2 inches, the pulley being 11.28 feet diameter, making 45 revo- 
lutions perminute. This gives v = 1592 feet per minute. The cross secLiou 

of the rope 3.14 so. inches. Hence A'o = . 0.0024. This 

5 1592 X 3-14 

is taken from an existing installation.* 

Example 2. In the jute mills at Gera the fly wheel of the engine is grooved 
for 30 ropes, ol 2.36" diameter, each rope transmitting 25 H. P. ; the velocity 
being 3000 feet per minute. This gives a specific capacity of N^ = 



3000 X 4-375 



: o.ooig. 



Example 3. The Berlin-Anhalt ZMachine Works has design rope transmis" 
sions in which ropes of i 18", 1.57", 1,97" diameter transmit forces, respec- 
tively, of 92.4, 165 and 264 pounds. The respective cross sections of the ropes 

N P 

are 1.09, 1.^3 and 3.04 square inches. Since = we have Nr. = 

V 33000 

P 
which gives in each of the three cases A'o = 0.0026. 



The cross section of the rim of a pulley for five ropes is 
shown in Fig. 880. For large steam engines the grooves are 

sometimes made on the 



fly wheel, such con- 
structions sometimes 
being very large and 
heavy. I 

The application of 
rope transmission in 
manufacturing estab- 
lishments simplifies the 
mechanism very ma- 
terially, since it enables 
the jack shaft and gear- 




FiG. 8S0. 

ing to be dispensed with. 

Such an arrangement is shown 



Fig. 88 1, in which five 




different lines of shafting are driven from one horizontal steam 
engine, sixteen hemp ropes beiug used in all. 

? 287. 
Sources of Loss in Hemp Rope Transmission. 

The use of hemp rope transmission reduces many losses 
which e.Niist in other methods and which materially reduce the 
efficiency ; the principal ones which need to be considered are 
the resistances due to journal friction, stiffness of ropes, and 
creep of ropes. 

a. Journal Friction. — In rope transmissions from steam en- 
gines the journal friction is usually great, because the large fly 
wheel requires journals of large diameter. The usual calcula- 
tions can only be given by indeterminate results, because the 
tension of the ropes sometimes acts with the weight of the 
other parts, and sometimes against it. 

If we consider the rope tensions T and t by themselves, as 
acting horizontally, we have from formula (100) the friction F 

= — f (,7" 4- {), which reduced to its corresponding resistance 

IK 

2 

to theroje, taking r ^2 ^^, gives a loss due to one shaft — 



-/ 



+ 1- 



we have : Ez 



-) \ — jy ) . If we take f =- 0.09 t and 
"ts, c 



double the result for both shafts, calling this combined loss Fz 
8 



X 0.09 X 4-; 



I which reduces to ; 



F,: 



2.R 



(272) 



Example I. In the first of the preceding examples we have also d = 6.3 
inches, and 2K = 135!^ inches, hence — — = 0.046 or a little over 4 per 

2/C 

cent. 



*See Zeitsclirift d. Verein deutscher Ingenieure, Vol. XXVIII, 18S4, p. 640. 



-{■See Engineer, Jan., 1S84, p. 38, for such a fly wheel 15 ft. fac^, 30 ft. dia.g 
weighing 140 tons, to transmit 4000 H. P. by 60 ropes. 
J See g 300. 



196 



THE CONSTRUCTOR. 



b. Stiffness of Ropes. — If we apply Eytelwein's formula (252) 
we have t? = j^ ( T-\- 1) taking both pulleys into consideration, 
and taking r ^ 273 and introducing T -\- t, gives Q =■ A\ P- 

It must be considered that the ropes are usually' quite slack, 
and that the co-efficient stiffness 6', may be taken somewhat 
less than Eytelwein's value. If we take -/^ as a fair approxi- 
mation, the ratio of loss is 

5 2 ^ \r- I 

= X 0.463- X4_ 

and calling this loss Es , we get : 



Es -= 



•33 



R 



(273) 



in which d is the diameter of the rope. 
Example 1. In the case of the preceding example, d = 



2", R = «7.75". 



This gives Ee = 1.33 



67-75 



= 0.07S or 7.S per cent. 



c. Creep of Ropes. — The loss through creep is more important 
in rope transmission than with belting (see I 2S4) and should 
not be neglected, although it cannot be so readily determined, 
owing to the division of the power among a number of ropes. 
It is practically impossible to insure a uniform tension upon a 
number of adjacent ropes, or to have them of exactly the same 
diameter, besides which the "working " diameters of the vari- 
■ous grooves differ slightly, so that additional slippage must oc- 
cur.* The resulting frictional loss is estimated by some at 
as much as 10 per cent., when the number of ropes is 20 to 30, 
and it is at all times important enough to be given considera- 
tion. The losses from stiffness and creep should be investi 
gated whenever practicable, as the resulting information would 
be of much technical value. 

Assuming the loss from creep in the case previously consid- 
ered to be 5 per cent., we have a total resistance of 4 4- 7-8 + 5 
= 16.8 per cent; which, since small values were taken in all 
cases, is not to be considered higher than the actual loss. 
This explains the numerous objections which have been raised 
(as in England) against the use of hemp rope transmission for 
very large powers (see J 301). 

?2S8. 

Pressure and We.-vr on Hemp Rope. 

As already seen, the surface of contact of the rope and pulley 
may be one of three kinds : upon a cylindrical pulley, in a 
semicircular groove, or in a wedge-shaped groove (Fig. S79), 
and to these f'ormula (241) can be applied. In case a, we can 
approximate b' as equal to ^ the circumference of the rope. 
This gives for the superficial pressure 



-. whence : 



'-'-^dR 



S R 



For case b, we have 4' = — d, whence 
2 



5 



i_d 
2 R 



(274) 



(275) 



In case c, the radial pressure Q, of the rope is divided into 
two forces Q' acting normal to the wedge surfaces and equal 

i Q 
to -^ — n— - in which d is the angle of the groove, and taking the 
sin if & £. I o 

contact surface on each side as \ the circumference of the rope, 
we have 



S 



sin a 



d_ 
~R 



which, for 8 = 30°, gives approximately ; 

P^ _ d^ 
S ~^ R 



(276) 



■^ The variation in adjacent ropes :nay be shown by pvitting 
coloring matter on the ropes and watching its distribntion . 



little 



Even under these unfavorable conditions the superficial pres- 
sure is not important, on account of the small value of S ; 
which, as already seen, is about 350 pounds. 



Example. — If .S = 350 pounds, and 



we have for a cylindri- 



cal pulley / = 350 X 2 X ~ = 2S lbs. for semicircular grooves, p = lb."? 

and for wedge grooves, when 6 ~ 30°, / = 56 lbs. per square inch. 

These low pressures cause but little wear upon the rope, 
hence the great durability of hemp transmission ropes, some- 
times extending to two or three vears ot use. 



£. 



^289. 
COTTON ROPE TRANSMISSION. 



Cotton rope is not so extensively used for purposes of trans- 
mission as hemp rope, although it possesses the advantages of 
great strength and flexibility ; the impediment to its use being 
its higher price. The application of cotton rope for driving 
spinning mule spindles, referred to [in l 265, is shown in Fig. 
8S2, in which T^ is the driving pulley and Z'j the driven pulley 




Fig. SS2. 

ou the carriage. This latter pulley is on the axis of a drum 7j 
from which light cords drive the spindles 7 ^. At L, L, are 
guide pulleys. The usual diameter of rope for 7j 7T, is 0.S6'', 
and for large machines with many spindles two such ropes are 
used, the pulleys being made with double grooves, these always 
being of semicircular section. 

On the ring spinning frame cotton rope of 0.4'' diameter is 
used on cone pulleys of 12 steps, giving changes of speed from 
3:1 to 2:3. The proportions of such pulley may be determined 
as shown in I 279, the grooves being semicircular. 

As already shown in ^ 265 cotton ropes have been used by 
Ramsbottom for driving traveling cranes. For this purpose 
ropes of i to J inch diameter are used, running at speeds of 
2500 to 3000 feet per minute, a weighted idler pulley keeping 
the rope taut. 

In view of the slow movement of the load, viz. : 20 to 40 feet 
per minute, it is questionable whether cotton rope transmission 
involving such a great transformation of speed, is advantageous, f 

C IP-IRE ROPE TRANSMISSION, 
i 290. 

Specific Capacity. Cross Section op Rope. 

In considering the transmission of power by means of wire 
rope the points to be determined are the cross section of the 
rope, and the deflection of the two portions of rope due to its 
weight. The cross section will first be considered by determin- 
ing the specific capacity (See ^ 2S0). This we get from (262) 



N,=- 



S, 



33000 



in which S-^ is the stress in driving half of the rope, considered 
either in connection with the driving or the driven pulley. 
The modulus of friction p is taken somewhat higher than for 
belting, since the angle of contact a is greater, and also because 
the co-efficieut of friction f for pulleys fitted with diagonal 
leather strips (see below) is very high ; early and recent tests 
giving_/= 0.22 to 0.25 and higher. The first value gives f/>» = 

2 

2 (See Fig. Si 6), and also the stress modulus r = -.=--. 2 

(See 239). This gives, in (262) if we neglect centrifugal force : 

I Si _ 5i 

33000 ■ 2 



iV„=- 



66000 



(277) 



t In some instances leather transmission ropes are used, formed of 
twisted thongs, these being used for light driving, as foot lathes, or 
light spindles. 



THE CONSTRUCTOR. 



197 



This gives high numerical values, which is also borne out in 
practice, since large powers are successfully transmitted with 
wire ropes of small diameter. It is good practice to take .S, 
for iron wire as high as 8500 pounds, and for steel wire up to 
20,000 pounds and even higher. This gives for the specific 
capacity, when : 
6'i = 2000, 4000, 6000, 8000, 10,000, 12,000, 14,000, 16,000, 18,000, 

20,000. 
N^ -.— 0.03, 0.06. 0.09, 0.121, 0.151, 0.182, 0.212, o 242, 0.273, °-3°3 
or approximatel}' : 

For Wrought Iron Wire JV^ = 0.03 to 0.121. 

For Steel Wire .... IV„ = 0.03 to 0.303. 
The cross section ^ is readily obtained, since N ^ q v N^ 
hence : 



Steel Wire. 



q = 66,000 



V Si 



(27S) 



We then have, if i is the number of wires in the rope, a diam- 
eter of work : f! = 2 — S'-. The speed v, of the rope may be as 

high as 6000 feet, but should not exceed this velocity on ac- 
count of the great stress upon the rim of the cast iron pulleys. 

I 291. 

Influence of Pulley Diameter. 

The bending of a rope about a pulley of a radius R produces 
a stress in each wire equal to 



■^i 



which, if we take both for iron and steel wire E = 28,400,000, 
gives : 



.J = 14,200,000 



J? 



(279) 



The driving half of the rope is therefore subject to a tension 
stress, both at the point of advancing and departing contact 
equal' to .S, + j in each wire. It is this sum which must be 
considered in determining the stress upon the material, and it 
must not be permitted to exceed the proper limits (See ^ 266). 
A practical upper limit for wrought iron wire is 25,000 pounds, 
while for steel it may be taken- much higher ; for hard drawn 
steel wire of good quality as high as 50,000 or even 60,000 
pounds. 

If we take as upper limits 25,000 lbs. for wrought iron and 
50,000 lbs. for steel wire, we have for the given values of S, the 

following values of 5 and of -^: 





Wrought Iron Wire- 







J? 


5 


5 


T 


7n 


24,885 


571 


1422 


24,174 


588 


2844 


22,752 


625 


4266 


21,330 


667 


5688 


19,908 


714 


7110 


18,486 


769 


8532 


17,064 


833 


9954 


15,642 


909 


11,376 


14,220 


1000 


12,798 


12,798 


iiii 


14,220 


",376 


1250 


15,642 


9954 


1429 


17,064 


8532 


1667 


18,486 


7110 


2000 


19,908 


5688 


2500 


21,330 


4266 


3333 


22,752 


2844 


5000 


24,174 


1422 


10,000 



5 


J 




1422 


49,770 


285 


2S44 


48,348 


294 


5688 


45,504 


313 


8532 


42,660 


334 


11,376 


39,816 


357 


14,220 


36.972 


385 


17,064 


34,128 


417 


19,908 


31,284 


455 


22,752 


28,440 


500 


25,596 


25,596 


551 


28,440 


22,752 


625 


31,284 


19,798 


718 


34,128 


17,064 


834 


36,972 


14,220 


1000 


39,816 


",376 


1250 


42,660 


8532 


1667 


45,504 


5688 


2500 


48,348 


2844 


5000 



If a still greater value of -r- is used for any given value of S^ 

than in the above table, the durability of the cable will be in- 
creased. The minimum pulley radius for any given sum of 

stresses 5i + 5 is obtained when — = .2, which in the tables 

gives for -r- = 833 and 417 respectively, as indicated by the 

full-faced figures. Even in this advantageous proportion the 
stress due to the bending of the wire around the pulley is double 
that due to the tension of the driving force. 

Example I.— Let jV= 60 H. P. K = 2052. The material is iron wire, .Si = 
8532 and the number of wires i = 36. We then have for the cross section of 
rope: 



: 66,000 ' 



60 



295= X 8532 



: 0.16 sq. in 



-V 



o.ib 



0.076" 



and the minimum pulley radius KsR = 833 X 0.076 = 63.3" or appoximately 
10 feet diameter. In order to obtain a velocity of 2952 feet per minute this 
requires about 93 revolutions. 
If we take S\ = s = 12,798 we find : 



q = 66,000 - 



63 



2952 X 12,798 



= o.ioS" 



fo.io 



whence 6 
R = IIII X 0.06 = 66.6" and n = 72. 

The question here arises, to what extent should the effect of 
centrifugal force be taken into account? If the velocity 
zi = 100 feet per second, with a stress 6' = 9000 lbs. we 
have from the first table in ^ 264 the value i — ;2 ^ 0.S7, so 
that instead of /a we have _/s' = 0.87 /a. Ufa = 0.22 we 
have/'a =; 0.87 x 0.22 x 'r = 0.70, and if y =, 0.22 we have 
/■'a = 0.S7 X 0.22 X 'T = 0.60. These give, by reference 
to the second table, in | 264, for the first value, the modulus of 

T 
friction p ^ —- 2.01, and for the second, p = 1.82 and a modu- 
lus of stress 
2 



2.22, which makes the specific capacity : 



— as great as previously obtained. This may be com- 
pensated for by making the cross section of the rope i.i times 
that obtained by the previous calculation. For lesser velocities 
up to 20C0 to 30CO feet per minute the effect of centrifugal 
force is much less and may safely be neglected, especially in 
the case of steel cables, in which much greater stresses are per- 
missible. 

Example 2. — How many horse power can be transmitted by a cable of 36 
wires, each 0.078'^ diameter; the velocity being 6500 feet per minute. We 

have iVn = —rz = 0.117 ; also q = (0,078) X 36 = 0.172 sq. in. 

II 66,000 4 



198 



THE CONSTRUCTOR. 



"VVe then have N = q v No = o 172 X 6500 X 0.117 = 130 H. P. For R we have 

r / ^ r' M.Z'^.OOO X 0078 , „ - „ 

from (275) K = ■ '— = 64.9" say 65 . 

17,064 

Hxainplc 3. — What would the horse power be if steel wire were used ? 
(See \ 266). ^1 = 17,000 lbs. and y^ will be twice = 0.234 whence N = 274 
H. P. If we ^desire durability of the cable we may make 5 only 28,440 in- 
stead of 34,128, and thus obtain R = — ' '— — '■ — = 3S.9 say only 40" 

28,440 

When the resistance P is directly given, which is rarely the 
case, we have from the relation g S^ ^ r P^ taking - = 2. 



P 

's,- 



(2S0) 



The maximum statical moment which may have to be over- 
come upou the driven shaft is sometimes given, as in the case 
of pumping machiuer}', etc. Dividing the precediug equation 
tiy (-79) we have 



14,200,000 [5 .S" 



. PJ^ 



and since o =: i — (5-, this reduces to : 
4 

(5 = 0.00564 V 4- V^j/'y? 



(281) 



and if we substitute for the moment P K the quotient of effect 

N 
from formula (135) P R = 63,020 — we get 



; = 0,251 V4- V 



S, 11 



(282) 



Example 4. — A pressure pump operated by a crauk on a shaft driven by 
Tvire rope, offers a resistance of 880 pounds, at a crank arm of 14.2 inches. 
This gives a maximum statical moment of P R . = 14.2 X SSo = 12,496 inch 
lbs. If we take i = 36 wires, and Si = 8500 pounds., we have from {2S1) : 



■- 0.00355 



vr^^ 



7,000 



12,496 = 



36 Af S500 

This gives from the table Ji = 833 )< 0.05 = 41.65, say 42". 
§ 292. 

Deflection of Wire Ropes. 

In order that the desired tensions T and / shall be attained 
in the two parts of a wire rope transmission, the deflections 
must be of predetermined values. The centre line of the rope 
will hang in a curve which lies between the catenarj' and the 
elastic line and which approximates closely to a parabola.* 

For the parameter c, of this parabola, we have for a deflection 
Ji, in a horizontal rope, Fig. SS3, 



S/i 



(2S3 



in which a is the distance between two points of suspension ; 
the deflection in the driving half being called //j, in the driven 



■^ The equation of the catenary is as follows 
a 







.(285) 



in which the tangential, vertical, 
and horizontal forces at a point 
xy may be designated as px^ ps and 
pc,p, being the weight for a unit of 

length, and 5" = v x- — c-. For 
the point of suspension this gives : 

K=p{h + c), \/=/ \/~h^~2h^, 

H = pc (286) 

in which the parameters is yet undetermined. In order to determine the 
latter let the equation of the curve be developed into the following series : 



^(..^ 



:.2.3.c3 ■ • 



+ I — 



■L.2C- 



Since the curve is always flat in rope transmission, the quotient -'— is a 

proper fraction, and both series are converging. Stopping at the third mem 
ber as giving sufi&cient accuracy we have : 



x=yzC 



(4^) 



= c + 



half, //2, and in the stationary rope ho. This gives for the tan- 
gential force A' at the point of suspension : 



K=p 



0+£) 



(284). 




X'- = X — c = ■ 



, which is the equation of a parabola. 



Fig. 883. 

All dimensions are to be taken in inches. For any cross sec- 
tion (7, we have A' = q S. and p ^ q ip \i ), in which ) is the 
■weight of the rope per cubic inch, and <p a coefficient, dependent 
upon the twist of the rope and upon the hemp core. The weight 
per cubic inch is y = 0.2S pounds, i/> is not always constant, but 
may be taken = |. These values give p ^ \ xo.28x<7 = 
o.3266§', and calling the coefficient 0.3266 := li, we have : 

From this we get : 

/,= ! A±/IZC^ ". . . (,88). 

2 li T 4 1//2 8 ^ ' 

I 

Since — ^ 3.061 we have, taking the negative sign. 

/, = i.53S-/^i.53^)-|;^ (2S9)- 

If we neglect the first member in the parenthesis in (287) we 
have for a close approximation : 

a'- 
h = 0.040S -^ (290). 

Example. Let a^ which we may take as the distance from centre to centre 
of pulley, be 262 feet, or 3144 inches; also let 5 in the driving side of the rope 
be 8500 pounds, and on the driven side 4250 pounds. We have from (289) 

K = 1.53 X S500 — J (i.S3 X S500)! — ?i|^" = 48" 

/'= = 1-53 X 4250 — J (1.53 X 4250)2 — -~^ = 95". 

The appro.xiniate formula (290) gives: 

h^ = 0,040s ^ = 47.45", and 

8500 

/io = O.O40S ^ = 94.89". 

4250 

The following method may be used to show the deflection h, 
graphically. The positive and negative signs before the radical 
sign in (288) indicate two values for /;, as will be seen in Fig. 8S4. 

The greater value is not of practical use, as it gives unstable 

S 

Uabil) equilibrium. Between the two lies a value h = V,. — , 

V' 
which is obtained when the quantitj' under the radical sign r^ 

(9, i.e. when 5 = — — . This we will call the "mean " deflec- 

V2 

tion and designate by hm. This deflection is important because 
with it the absolute minimum stress exists in the rope (see note 
at the end of this section) ; and this stress, which occurs with 
the deflection hm, will be designated Sm, and is : 

Sm = li — >— (291). 

V2 

or for the preceding value V = 0.3266, 

Sm = (292). 

4-33 

c 
and since hm = J'a . — tj we have for the mean deflection 



THE CONSTRUCTOR. 



199 



hm = ■ 



Vs 

Dividing (2S8) by {293) we have, after some reductions : 

tint K^m 



(293). 



(294). 




From this we obtain the following geometrical construction 
of Fig. 8S5. With a diameter = ^i''. describe the semi-circle 
1.2.3, and join the point 3 of the quadrant 2 . 3 with 2, or i ; 

a 

2 d 

then 3.2^3.1 = — ■— =: — — = the mean deflection /hu. 
V 2 v8 




Fig. 8S5. 

Lay ofrthis distance perpendicular to i . 2 at 2 . 4, and on any 
scale (not too small) lay off from 2 to 5, the stress Sui. deter- 
mined from (292). From 5 lay off, on the same scale, 5 . 6, equal 
to the given stress S, and from 6 draw the arc 5 . 7. This gives 

2. 7 =6. 5 — 6.2=6.5— n/(6 . 5)^^- (5'. 2)-, which is = 5 — 
"v .S^ — Sm''. If we now draw 4 . S parallel to 5 . 7 we have 

2.8 2.7 A 
2 . 4~ 2 . 5 



-, and hence 2 . 8 = /;, 



The value /lo of the stationary rope is that of a parabola of a 
length midway between those forZ/i and //^ and is equal to : 



/to 



— — -- = 0.67/;,, + 0.2S//1 . 



(29s) 



_ It may readily be constructed graphically from the first expres- 
sion. It is not essential that the driving part of the rope should be 
the upper portion, as the lower part may drive, as in Fig. 886. The 




Fig. SS6. 

ropes will not touch, when stationary, if //, — //j -c 2J?. Owing 
to the fluctuations due to the action of wind, or of sudden 
changes of load, the minimum distance should not be too small, 
and is best kept greater than 20 to 24 inches. 



Note.— We have from (287) : 



rf 5 = V j d/i + }s\o — -p; ] dh I which gives for the mini- 



"] 



mum of J> ; 



L ^ ~ V ' ^ l^J J ■ "^"* Wi ^^ ^^^ paramet. 



er, or C, 



Wj. 



hence = 1 — -— or Cm = hm = — — and hence from (2S7) : 

llm v^g 



, = .,.( 



hm -t- 



Vln 



=.r- 



L-x/S 



+ —7^ + V' 



n/8~ 



a 



In Fig. SS4 is shown graphically how for each value of//, the 
parameter c can be found, by constructing the proportion 
a 

— = — . In the figure, 2 . 5' =: 2 . 4' == //' ,• also 4' . 6' = — ; 
ci c 2 

2 

and 6' . 7' perpendicular to 6' . 5' gives at \' . 7' the parameter 



c'. In like manner : 2 . 5' 



2.4' 



h'^ ; also A," . 6" = 



2 



and 6" . 7" perpendicular to 6" . 5" gives the parameter 



4" . 7" = c'' 

To determine the vertex 4" of the lower parabola we have : 



h' + -^. = h" + -^— whence h' — h" ■- 



8 \h" h' J 



a} h' h" d- 

^= -^ — T , , „ ■ This gives A' /;/' = -3- which as shown above 

= hm-- 

If, as before, we make 2.8^ hm, and 2.5'^ //' and 
draw through 8 a normal to 8.5' the normal will intersect 
2 . 4" at 4" which is the desired ape.>i. The lines 5 . 6, 5' . 6', 
5" . 6'^ intersect each other at the middle of the half-chord of 
the parabola at 9. This may be used in the construction by 
draw'iug from 9, the line 9 . 6, 9 . 6', 9 . 6", and the correspond- 
ing normals give the parameter points 7, "', 7". The directrix 
of the parabola lies at a point distant )4 c from the vertex. For 
the mean parabola the directrix is Lm, midwa}' between 4 and 7, 
and the focus Fm is at the middle of hm, and is also the centre 
of the circle 5.6.7. i. 

In the figure is also shown another curve which indicates the 
values of S. The proportional value of // from formula (2S7) 
taken from the line 2 . 11, shows that h is in inverse proportion 
to the hyperbolic line 10' . 10 . 10". The ordiuates of the hyper- 
bola, taken from the axis of abscissas 2 . 7 gives the values of 6" 
for the corresponding values of //. The ordinates 4' . 10' and 
4" . 10" give the equal stresses S' and S", and 4 . 10 the mini- 
mum stress Syn The dotted hyperbola on the upper right, gives 
the corresponding thrusts in a parabolic arch, and the curve in 
an arch corresponding to the catenary is the line of thrust. In 
this also we find the mean height the most economical, the 
lower ones being stable, and the higher in an unstable equilib- 
rium, dependent upon the thickness of arch ring and distribu- 
tion of load for their stability. 



200 



THE CONSTRUCTOR. 



\ 293- 

TIGHTENED DRIVING ROPES. 

The deflection of transmission cables often becomes incon- 
veniently great. In many cases, however, it is possible to 
reduce its amount by increasiug the tension to a greater extent 
than is necessary to prevent slippage. This requires the cable 
to be made correspondingly stronger in order to resist the in- 
creased tension. The modification in the preceding discussion 
of forces and dimensions is here given, the various terms being 
given the subscript j to distinguish them, {Ts, is, Ss, <5s, instead 
of T, t, S, i). The tension T, as shown in I 290, should not be 



c ^ , J. I / > r, 14,200,000 X 0.026 ^, . , 

12 feet. According to (279) R = = 32.45, say 32J4 inches, 

11,376 

which gives Ao — Aj ^ 2 7? and the driving part of the cable must be. above. 
The above result shows that the centre of the pulleys must be more than 
R -t- h., or 24ft + 2^' 8^^" above the ground in order to clear. To reduce this 
height we must tighten the cable. Suppose we made the diameter of the 

wires = 0.04" instead of 0,024". This gives -r- 1.67, and from the table, col- 

o 

umns 4 and 6, line 11, .Si's = 0.89, 5 = 12,650, and hence we have tna = 

162 — 144 = iS". 



0.0408 - — -J— =162", and his — h 
12,650 



We also have R = 



11,376 



X 0.04 == 50". These values give hi. 




^^^^sm//,^mm7:^.i^M }m-^m^^m^-7^^:^^M 



less than iP, and if this is increased by a given factor m, we 
have is ^ Ts — P, and also : 



Ts = m T= 2m P, ^ 
is ={2m—i)P, 

is 2111 — I 

Ts 2111 



(296). 



In order that the stress ^i in the driving part shall not be 
changed we have for the stress in the driving part, instead of 

S, 



— ^, the following : 
2 



2m 



(297)- 



The diameter 6 of the wire, if calculated from (280) is modi- 
fied to 

<!^ = 6>/m {298). 

or if taken from (281) or (282), we take 

5., =(i\^'^;77 (299). 

from which the following table has been calculated. Tightened 
cables are frequently applicable where moderate powers are to 
be transmitted. 

TABLE FOR TIGHTENED CABLES. 



Ts 


n 


ts is S„s 




is 


^s .3/ 










-f--^1n 




m y. 


p 


t P S., 


Si Ts 


& "' 


1.6 


3-2 


2.2 


0.69 


1.26 


1. 17 


I.? 


3b 


2.6 


0.72 


1-34 


1.22 


2.0 


4.0 


3.0 


0.75 


1.41 


1.26 


2.2 


4-4 


3-S 


0.77 


1.48 


1.30 


2.4 


4-8 


3-8 


0.79 


1-55 


1-34 


2.6 


S-2 


4.2 


0.81 


161 


1.38 


2.8 


5.6 


46 


0.82 


1.67 


1.41 


3-0 


6.0 


•5-0 


0.S3 


1-73 


1.44 


3-2 


b.4 


5-4 


0.84 


1.79 


1.47 


34 


6.8 


5-8 


0.85 


1. 84 


■•50 


3-6 


7.2 


6.2 


0.86 


1.90 


I-S3 


3.8 


7.b 


6.6 . 


0.S7 


1-95 


1.56 


4.0 


8.0 


7.0 


0.8S 


2.00 


1-59 


4.2 


8.4 


74 


o.SS 


2.05 


1.61 


4.4 


8.S 


7.8 


•0.S9 


2.10 


1.64 


4.6 


Q.2 


8.2 


0.89 


2.14 


1.66 


4.8 


9.6 


8.6 


0.90 


2.19 


1.69 


50 


1 0.0 


9.0 


0.90 


2.24 


1. 71 



Example. — Given, A'= 5 . 5, ?i = 100, a = 590.4 ft. = 7086 in. It is required 
to cover this distance with a &ing"le stretch of cable. If we take Si = 14,220 



lbs., and j- = 11,376 lbs., we have — ^ — X ■ 



".376 .s ■ 5 
14,220 100 



0.044. If 



; = 36 we have from {282) 5 = 0.251 ^ — — /Vf c.044 =0.024 inches. We then 
get from (290) : /n = 0.040S ~ = 144" = 12 feet, A^ = 24 feet, and 7^2 — Ai = 



<^ 2j? and we may therefore place the driving; side below without danger of 
interference. The greatest deflection occurs when the cable is at rest, and 
from (295) we have /los = i^q inches, and the total height for the pulley cen- 
tres is 149 + 50 = 199" or 16" 7". This example is shown in Fig. 887, in which 
the dimensions, however, are in the metric system. 

? 294. 
SHORT SPAN CABLE TRANSMISSIONS. 

When the distance between pulleys is short the deflection 
must not be too small if good results are to be expected. To 
this end a small value should be taken for S^, and heuce the de- 
flection is first to be chosen and the corresponding value deter- 
mined from (287) which is readily done. For moderate powers 
wire rope transmission may be used in this way for short spans 
very sixccessfully. 

Example.— t,^t iV= 5 horse power, to be transmitted over a span of 65.6 ft., 
or 787 4 inches; the number of revolutions to be 150, and the deflection 40 



inches. We have from (2S7) 5j=o.3266 (40 + - 



J = 645 lbs. Taking iron 



40 X S , 

wire, and making 5 + -r = 25,600 lbs. We have .s-= 25,600— 645 = 24,955. If 
we make the number of wires /= 36 we have from (282) 



^36 Ar 645 J50 



We then have from {279) 
R = 



and V = 



14,2000.000 X 00? 3 
25,600 

150 X 2 TT X 46 



= 45-9 say 46", 



: 3600 feet, 



all of wliich values are quite practicable. 

? 295. 

TR.iNSMISSION WITH INCLINED CABLE. 

A transmission at which the pullej-s are placed at different 
heights is called an inclined transmission, and the curve in such 
a case is uns}'nimetrical. For a given distance a, between the 
verticals through the ends of the curve, and for a difference in 
height //, we have for the deflections /?/^.-r, and /;/' =x,, Fig. 
SSS, and for the ordinates j'l and y.^ of the two branches of the 
curve: „2 ^ ^2 //- ~ 

A-i = A' = — + — — _ — 



and 




a H 








a , H 




in which the parameter c is yet unknown.* 





(300) 



(301) 



* Deduced as follows ; jj-- = 2 c x\^ y^ = 2 c xn, y^^ y'y=a, x» — xi=H^ whence; 
y-r — y-i- = 2 c {x2~ xi) = 2c H, i.e. {y.. -\-y\) {yi—yi) = 2<://andhencej'o— j^i 

=: 2 c , etc. 



THE CONSTRUCTOR. 



201 



For the parameter c, we have from (286) I\^p{h-\-c) or 
Sq = '^ q(Ji -\- c) and if we consider the lower pulley as bearing 



The deflection is : 



39)6° 



I X 12,962 



12,962 



X 0.0025 — 98.5=67.1", and /;""= ^2' + 197=264.1, 



whence 




y\ = 1968 — 0.05X 12,962- 



■■ 1319.9. 

The stress on the rope, instead of being exactly 8500 and 4250 pounds, will 
be, according to (304) : ■ 

8500 -t- 0.3266 X 197 = ^564 lbs., and 

4250 + 0.3266 X 197 = 4314 lbs. respectively. 



-ioora- 



f IG. SS8. 

S' 
the lighter load we have : S' = ip (c + .r,) whence f = -t ^j. 

Substituting the value of Xi, from (300) we obtain after reduc- 
tion. 



-^2— (302) 





Fig. S89. 

The arrangement is shown in Fig. 8S9. the vertical dimensions being three 
times the scale of the horizontal, and all dimensions being in metres. 

Example 2. — Suppose the distance a = 3936 inches, and S\ = 8500, and S 



The plus sign before the radical indicates thatwe have chosen 
the "stabil" parabola (see Fig. 8S4), and hence obtain the 
greater of the two values for the parameter. The parameter 
thus being determined, we have x\ and }\ from (300) and (301). 
For the upper branch of the curve the stress S" is to be de- 
termined at the upper pulley. We then have S" = 'Z' (i^ + A'2). 
Subtracting from this 5' :=i/' {c -\- x-^ we have 

S" = S' ^--^ {x.^ — x,)=S' -{-^ H (303) 

and if 1/'^ 0.3266 we get : 

S" = 5' + 0.3266 H (304) 

ExampU I —Let a = 32S felt = 3936", S' = 8500 lbs. If //= O, we have from 
S500 



4250, as before, but the vertical distance H = 1968", or - 
iig Side : 

J- + 9S4 ^ '' ^ 



We then have 



{a) For the Driving- Side 

8500 



2()6 




= 23361, whence 



\ X 23361 

andjj'i = 1968 — 23361 X 0.5 ^ — 971- inches, 
the minus sign indicating; *>hit the apex of the parabola lies without the 
space between the pulleys. 

{b) For the Driven Side : 
4250 



o 3266 



+ s 



! +0.5= 



<302) C = 



0.3266 



- + 



/ ^ 8500 ^ - 



■+ 
39362 



V" 



o 3266 ^ ^ ^ ^ 



3936- 



■ + 0.25 X 



2 4- 0.5- 
12271 



= 25,95i(inches. 



8 X 12271 
andj'i = ig68 — 12271 X 0.5 = 
and the apes again lies outside. 



i+o 125) 

= 708 inches, 
4167 inches. 



12271, whence 




Fig. S90. 



This value in (300) gives xi= x-i- 



=/,, = - 



3036' 



8c S X 25,951 

For the slack half of the rope we have 5'o = 4250, and 
4250 
0.3266 



■= 74.62". 



■ + 



/(i 



The general arrangement is shown in Fig. S90 all d mensions being given 
in the metric system, and the vertical and horizontal scales being the same. 
The increase in the .stresses is more marked than in the previous example, 
on account of the increase in the value of H. We have S\ = 8500 + 0.3266 X 
1968 = 9142 lbs., and S{ = 4250 + 0.3266 X 1968 = 4892 lbs. 



3266' ) 3936- 

— y 8 



whence h^ 



3936" 
'■ 8X 12,86 



We then have : 



= 150-5 '. 
Suppose now that H ~ o 05 a - 

[a) For the Driving^Side 

8500 

c= + 

2 +0.05 - X 2 + 0.05: 

■which is slightly greater than when H^O. 

39362 _ 26,018 



\f f ^6 + ^'■A '_ 3936' 

' V 2 + 0.05= y 8(1+0.00125) 




We have also, from (300) h'l = 
and /^i" = /ii + 197" = 205.45". 
The distance .j'l then becomes: 

.3936 



8 X 26,018 



+ 



X 0.0025- 



8.45" 



j)'l = -'- — 005X 26,o;S = 
(i) For the Driven Side : 



667.1, 



.=iSi!^+ 4/(11^+93. y__^,^^ 

3-Ho.o5= + ▼ V 3 .,■ 0.052 y 8(1+0.00125) '''°'- 




Fig. 891. 
The upper limit for this form of rope transmission is that ia 



202 



THE CONSTRUCTOR. 



■which the parts of the rope are vertical, in which case the par- 
ameter ^= cxi. In this arrangement the necessary tension must 
be obtained by the use of weights, spring, or the like. B3' using 
guide pulleys, a combination of horizoutal and vertical trans- 
missions may be made, as in Fig. S91, and the tension obtained 
by the deflection in the horizontal part. 

§296. 
Construction of thk Rope Curve. 
We have considered the curve as an ordinary parabola. 




Fig. S92. 

When the apex C, Fig. S92, has been determined, bisect the 
two parts J?! Cand Z?j C"of the horizontal tangent Bi D^, at Q 
and C, join B C^ and D C,, and these two lines will give the 
direction of tangents to the curve at the points of suspension B 

and D. Then divide C Cj into equal parts C, i, 2, 3 and 

C, B into the same number of equal parts C, I, II, III , 

and by joining these points we obtain a number of tan- 
gents which include the curve. The other portion C C, D, of 
the curve is constructed in a similar manner. When the apes 
of the parabola falls beyond the lower pulley, only one portion 
of the curve is used. 

? 297. ■ 

Arrangement op Pulleys. 

When the transmission pullej'S are far apart, and not high 
above ground, supporting pulleys must be used for the rope. 
In some instances this is only necessary for the driven part of 
the rope, the driving part being left unsupported, as in Fig. 893. 




'^'W'lMr^Wt^^.twy^iWKS; 



/I. I 

! i 



Fig. 893. 



Each portion of rope between two pulleys may be called a 
"stretch" of rope, so that in the above instance we have the 
driving part in one stretch and the driven part in two stretches. 
If it is necessary to support both parts it is often practicable to 
use half as many supporting pulleys for the driving part of the 
rope as for the driven part as in Fig. S94. 




Fig. S94. 

These pulleys are called guide pulleys to distinguish them 
from the main transmitting pulleys and their supporting struc- 
tures are called supporting stations. 

Another arrangement has beeu used by Ziegler, as shown in 
Fig. S95. 




Fig. 895. 



This consists of a number of shorter transmissions, using 
double grooved pulleys, or two single grooved pulleys at each 
station. In this arrangement it is advisable to make the 
stretches of equal length so that a single reserve cable will 
answer to replace any one which may give out. 

It is always desirable to run a transmission in a straight line, 



and especial care must be taken to have the successive pulleys 
all in the same vertical plane. If it is impracticable to run the 
entire distance in a straight line it is necessary to introduce 
angle stations. These may be constructed as in Fig. 896 a^ 





Fig. S96. 

using vertical and horizontal guide pulleys, but this requires 
six pulleys, three for each part of the rope. A simpler arrange- 
ment is shown at Fig. S96 b, two pulleys and a pair of bevel 
gears being used. 

In many cases it is desirable to take off a portion of the power 
at intermediate stations either by shafting or rope transmission, 
and this may readily be done by a variety of arrangements of 
gearing and shafting. 

It is most important that the pulle)-s both for supporting and 
transmission should be amply large in diameter. Many rope 
transmissions have worn out rapidly, simply because the diam- 
eter of the pulleys has been too small. The intermediate pulleys 
for the driving side ought to be the same size as the main driv- 
ing pulley in order that the total stress S-)- j (see § 291) shall 
not be greater in the former case than in the latter. The sup- 
porting pulley for the driven side may be smaller because the 
stress S, is smaller generally, being ji, S,, or for tightened 
transmissions (I 2S9) being equal to [2m — 1) 2111 Sy The 
smallest permissible pulleys may be determined from formula 
(279) and the table of ^ 291. 

Example i.— In an ordinary wire rope transmission let 5i = 8500, Sn = 
4250, and the wire of wrought iron, 5 being = 0.06". From the table in g 291 
we have for the mininiinu radius of pulley, R = 833 X 0.06 = 50" or 8^t 4" dia. 
and for the supporting pulleys : R.-^ = 667 X 0.06 = 40 or 6" S" dia. 

Exajnple^, — Let 6= 0.04. S[= 568S, 5^ = 2S44, and for iron wire we have 
R = 2S.56 say 30", R.2 = is"- 

Exajnple 3.— In a'tightened transmission let m = 3, and 6 = 0.06", Sx = 

S500. Sn =5i 



■Jill}. \ = i 5j = 7080. R = 50" as before, and /?» = 769 X 

2 X 3 o 

0.06 = 46", a difference which is hardly great enough to be of practical im- 
portance. 

I 298. 

The Construction of Rope Pulleys. 

The low value of the coefficient of friction of iron on iron 
makes it impracticable to run the wire cables directly upon the 
bare metal rim of the pulley, and hence various attempts were 
early made to fit the groove of the pulley Ivith some soft material. 
After early experiments with wooden rims fitted with leather, 
or rubber, it was practically shown that turned iron rims fitted 
with leather filling placed edgewise in the bottom of the groove 
gave the best results. "^ 




l0+0,6fl 




sjfl.sd 



Fig, 



In Fig. S97, is shown at a, a rim for a single pulley and at b, 
for a double one, both being of cast iron. The proportions are 
given in terms of the diameter d, of the cable, and in the illus- 
trations the constants in the various proportions are in milli- 
meters. The sides of the grooves are made at an angle of ■yf 
with the plane of the pulley in the case of the single groove 



* See D. H. Ziegler, " Erfahrungs resultate iiber Betrieb und Instandhal- 
tung des Drahtseiltriebs." Winterthur, 1S71. 



THE CONSTRUCTOR. 



203 



pulley, but this gives au excessivel}' heavy middle rib for the 
double pulley, and hence the inner angles are made 15° as 
shown. The smallest diameter of rope for practical use is 
d = 0.04". The superficial pressure />, may be calculated from 

d 
(274). If, for example, i = 36, we have from (244) — - =^ S, and 



if 



J? 



1000 and S = S500, we have ; 



8(5 



p = 2S , = 136 lbs. per sq. in. 

lOOOfi 

a pressure readily borne by the leather filling. 

The bottom grooves are made with a dovetail bevel in order 
to keep the filling from being thrown out by centrifugal force. 
The filling of leather may be made of pieces of old belting 
placed on edge and forced by driving into the dovetail groove; 
if new leather is used it should be softened by soaking in train 
oil. Rope sheaves for hoisting machinery, which are only used 
for guiding and supporting the rope, were formerly used with- 
out any filling, the rope resting on the bare metal. It is be- 
coming more and more the practice to use a filling in the bot- 
tom of the grooves of such pulleys, vulcanized rubber giving 
good results. 




Fig. 89S. 

The construction of the rim of Fowler's " Clamp Pulley," re- 
ferred to in Fig. 794 c, is shown in Fig. S9S a, the clamps being 
pivoted to blocks by means of bolts with anchor-shaped heads. 
The pressure upon the rope is the same as in the case of a wedge 
groove of equal angle, and the pulley as made by Fowler, 
has one clamp ring mounted upon a screw thread cut upon the 
pulley, thus enabling adjustment to be made for wear upon the 
clamps and for the reduction in the diameter of the rope. Fig. 
S9S b shows an American form of clamp pulley, somewhat sim- 
pler in construction than Fowler's. The clamps are pivoted on 
half-journals (see § 95) and the angle is not so small as in the 
preceding form. 

The arms of rope pulleys are usually made of cast iron as well 
as the rim, although the intermediate supporting pulleys are 
sometimes made with wrought iron arms, as in Fig. 901. Large 
pulleys, when of cast iron, are usually made in halves, for con- 
venience of transportation. 

The number of arms A, may be obtained from : 



40 d 



(305) 



Cast iron arms may be either oval or cruciform in cross sec- 
tion, and the width of arm //, in the plane of the pulley, if pro- 
longed to the centre is : 



: 4^ ■ 



~A 



(3°6) 



For arms of cruciform section, the thickness of the arms e 
may be made i //, and the rib thickness e' = 73 e. Arms of oval 
section may be made of the same proportions as for belt pulleys, 
the thickness being made one-half/; at all points and the width 
at the rim being 7-^ /;. 

Arms of cruciform section are usually made straight as at (7, 
Fig. 899, but arms of oval section are frequently made curved 
as at 5. 

To draw the curved arm make the circle O A of a radius = 
}2 J? and divide it into spaces for the desired number of arms. 
Make A E = ^<, A B, and draw O Cnormal to ^ O and (Twill 
be the centre for half the arm, and the other centre will be at 
D, the radius D £ being equal to C E. 

When straight arms are used the hub should be divided as in 



Fig. 899 a, in order to avoid injurious stresses from shrinkage 
in casting. The spaces are afterwards filled in with fitted pieces 
of iron and a ring shrunk on each side to hold all together. 
The proportions of hubs are the same as in \ 283. 




Fig. S99. 

The distance between journals for the intermediate and sup-" 
porting pulleys varies from \ R io J R. The load upon the 
bearings consists of the sum of the weight of the pulley and the 
vertical component of the various forces upon the rope, and 
this can best be determined graphically as shown in Fig. 900. 




The weight G, of the pulley is so dependent upon slight 
variations in the thickness and section of rim and arms that a 
general formula of practical value cannot be given. The follow- 
ing examples from practice are given : 

Example I. — In an executed transmission by Rieter & Co., at Oberursel, 
near Frankfurt a. M., the pulleys are made with twelve straight arms of 
oval section and are 12 ft. 3.6 in. diameter. The main driving pulleys at the 
end ol the transmission, with single groove, each weigh 2525 lbs. and the 
intermediate supporting pulleys, with double groove, each "weigh 2780 lbs. 
The rope is made of 36 wires, each being 0.07" diameter. 

Exauiple 2. — The Berlin-Anhalt Machine Works Company nmkes a line of 
rope pulleys with wrought iron arms as in Fig. 901, tlie weights being as 
follows : 

R = 20" 24" 2S" 32" 36" 40" 50" 60" 70" 

G = 176 211 24S-30S 2S1-343 316-3S7 316-506 528-570 74S g6S 

In these instances the weight upon the bearings is not great. 
The journals for these pulleys should be made long, in order to 
reduce the superficial pressure, and swivel bearings wit'u cast 
iron boxes (J/, 116) can be used, which with self-oiling devices 
will give good service. In many cases the journals are made of 
hardened steel in order to combine the greatest security with 
the minimum size. 

Example 3. — The intermediate pulleys in Example 1, give a total pressure, 
according to Fig. 900 h, upon the bearings, of 3036 pounds, or 1518 pounds on 
each journal. If we make /= 1.5 d according to the table in §91 we get for 
d only ii^ in. In the actual case, however, the journals are 3->4 in. diam- 
eter, giving a greatly reduced superficial pressure and thus insuring the 
most complete lubrication. In this case we have the actual length / = 4.7, 

whence P = . = 86 lbs. per sq. in. If, in order to use formula (80) 

3-75 X 4-7 

we take — - = // and make 5" = S500 as before we have : 
a 



-=Nf^^V-^- = 



and =4^=8''', This gives ; 
1518 



P='^ 



X8 



1,9" say 2" 
= 95 lbs. per sq. in. 



204 



THE CONSTRUCTOR. 



which is such a low value that even half boxes, similar to those m Figs. 324- 
325 could be used. Bv using hard steel bearings even this small fnctional 
lesistance could be reduced to J4 the amount due to the above dimensions. 

The pulleys for rope transmission should be most carefully 
balanced, as any vibration causes serious oscillation of the rope 

? 299. 

Construction op the Puxley St.vtions. 

The extraordinarily high specific capacity of wire rope tran.s- 
mission has, as already said, caused it to be used especially for 



upon which the pulleys are carried. * The following are exam- 
ples of well designed and constructed stations. 

Fig. go I shows a design for an intermediate station of masonrj'. 
The foundation is of rough stone-work and the superstructure 
of brick-work. 

Stations similar to this are used in the transmission at Ober- 
ursel, referred to in the preceding section, and erected in 185S. 
This installation is used to trausmit 104 horse power over a dis- 





FiG. 901. 



the long-distance transmission of power. It has been found 
particularly adapted for the transmission of the power of natural 
falls of water to places where it can be utilized and has thus 



Fig. 903. 

tance of 3168 feet (966 meters) divided into eight stretches, giv- 
ing two terminal and seven intermediate stations. Each 
stretch = ^'/^ = 396 feet long ; /? = 74 inches, « = 1 14.5, z' = 





III II IK 11 h yiiiiiiiniiPLj_ 








Fig. 902. 

materially advanced the use of natural sources of power. In 
such transmissions one of the most importaut and difficult por- 
tions of the work consists in the construction of the stations 



Fig. 904. 

4400 ft , (' — 0.07", i = 36. The difference in level between the 
two terminal stations in this case is 145 feet. 

The transmission of the water power from Scbaffhausen, con- 



* These have been fullv discussed in a work by D. H. Ziegler treating ot 
the installations made by Joh. Jak. Rieter, Winterthur, 1S76, and printed 
privately. 



THE CONSTRUCTOR. 



205 



structed by J. J. Rieter & Co. and iu operation since 1 866, is 
used to transmit a total of 760 horse power developed by the 
Falls of the Rhine. Of this 200 horse power is transmitted 
direct to the left bank by means of shafting ; 560 horse power 
is carried across the Rhine in one stretch, the distance a being 
3S5 feet, using two similar ropes carrying 530 horse power. 



(« = i8o, j? = 88,;<' 



-4636 ft.) and a third single rope carrying 



30 horse power (;; = 180, i'? = 35.4"). Of this power there is 
about 4S0 horse power transmitted over three principal stretches 
of37S, 332, and455feet. Thenumberof wires in the heavier cables 
is i =: So, the thickness of wire <5 = 0.074", 'be rope being made in 
8 strands of 10 wires each. One of the intermediate pullej' sta- 
tions is shown iu Fig. 902, and this is an excellent example of 
good style in construction. In this case there are two pulleys, 
side by side. There is a guard shown over the pulleys, to pre- 




vent possible jumping of the cables out of the grooves in the 
pulleys, but this has been omitted in later instances as un- 
necessary., 

Messrs. Rieter & Co. have also installed a system of turbines 
and rope transmission at Freiburg, for the Societe generale 
Suisse des eaitx /orHs, of which 300 horse power is in a long- 
distance transmission. The power is carried in five stretches of 
502 feet each, to a saw mill, the difference in level being 26S feet. 
Cue of the stations with two supporting pulleys is shown in 
Fig- 903. this one being quite high ; a similar station, No. II, is 
placed in a tunnel, through which the rope passes. The num- 



ber of wires i = 90, the diameter of wire d = 0.072", the cable 
being made in 10 strands of 9 wires each, /? = 88.6", n = 81, 
'' = 3743 ft. From this point the power is divided by an angle 
station and one part is delivered to the saw mill and balance 
transmitted to a number of minor establishments. 

An angle station is shown is Fig. 904, and this form is also 
used when a portion of the power is to be taken off. 

A fourth large installation of turbines and rope transmission has 
been executed by the firm of Rieter & Co., for the Conipagiiie 
giiierale de Bellegarde, at the latter place, for the utilization of 
the well-known Perle dii Rhone. The combined power of the 
Rhone and the Valserine is exerted upon five turbines of 630 horse 
power each, giving a total of 3150 horse power which is trans- 
mitted by cable to the Plateau of Bellegarde.* 

At Zurich, the city has utilized the power of the Limmat by 
means of turbines and rope transmission built by the firm of 
Escher, Wyss & Co. In this case the stations, which for various 
reasons are quite high, are made of wrought iron, as shown in 
Fig. 905. The entire installation develops 11 50 horse power, 
of which 750 horse power is used for the city water works. 

At St. Petersburg a rope 
transmission in ten stretches 
is used to drive the Imperial 
Powder Works, the power 
being delivered into the 
buildings bj' shafting from 
each of the ten stations. 

Amodificatiou of Herland's 
device for putting on belts, 
has been made by Ziegler for 
the purpose of putting the 
wire cables upon the pulleys. 
As shown in Fig. 906, it 
consists of a curved piece of 
angle iron, clamped tempo- 
rarily to the arm of the 
pulley in such a manner as 
to lead the rope into the 
groove of the pulley. The 
short radius to which the 
I 




Fig. 906. 




Fig. 907. 



rope is thus once bent does not 
appear to have an injurious effect. 

When a transmission rope is car- 
ried over a public or private road 
a guard should be used as a pro- 
tection in case of breakage of the 
rope. A simple form used by Rieter 
& Co. is shown in Fig. 907, and 
consists of a sheet iron trough about 
18 inches deep and ten feet wide, 
carried by two stationary suspen- 
sion cables is H H. 

? 300- 

Efficiency of Rope Tr.\xsmissiox. 

The injurious resistances iu wire rope transmission are mainly 
those due to journal friction and stiffness of the rope ; the slip 
and the atmospheric resistance of the pulley arms being in- 
significant, t 

a) Journal Friction. — We have from formula (100), F^= — fO, 

in which Q is the load upon the journal. For a circumferential 
speed c, at the journal, we have a resistance in foot pounds : 



Fc = 



fndO 



(307) 



Example I. — In the case of the transmission at Oberursel a number of ex- 
perimental determinations were made. For a pair of journals Q = 2948 Ihs., 
d = 3.75" and «= 114.6. For a coefficient of frictiony = 0.09 (experimentally 
determined) we have ; 



37«5S 



0.09 X 1 14-6 XjvTSX^JmS 



1. 14 orse power. 



: 37,658 foot lbs. 



This gives for 8 iations a total loss of 8 X 1.14 = 912 horse power. The 
maximum power iransmitted is 104 H. P. and the minimum 40.3 H. P., so 
that this gives a less of about 9 per cent, of the maximum and 22 per cent, 
of the minimum. This shows the objection to the use of too largcjournals. 



*See Engineerint^^ Vol. 37, 1874. 
t See Leloutre. 



2o6 



THE CONSTRUCTOR. 



b) Stiffness of Rot>e.—\Jsmg Weisbach's formula (253) given 
in ?i 268"! 



6- 



1.07S + 0.093-^- 



(3°«) 



we have, calling T' the tension on the rope : 

& = 0.093 r- (^'1-6+ ^J • • • 

for the resistance in foot pounds. 

Example 2.— In the preceding case, v — 440° ft-, -f = 73-S", and T' = % 
(T'-f /) = 0.5 X 202S = 1014 lbs., whence : 



Su = 0.093 X 4400 [ 



73 » 



J036S ft. lbs. 



This resistance comes twice at each station, and for eight stations we have a 
total of 2 X 3 X 10.36S = '63,888 foot lbs., or nearly 5 horse power. Adding 
to this the journal resistance we have a total of 9.12 + 5 = 14. 12 H. P. The 
direct measurements of Ziesrler gave '3-341 H. P , which is a reasonably close 
verification of the calculations. The total loss of efficiency is therefore : 



104 

.14-12 

403 



13.6 per cent, of the maximum. 



: 35 per cent, of the minimum, 



the lesser of these being a very excellent result. 

i 301. 

ReuIvEux's System op Rope Transmission. 

In the preceding sections the utility and importance of wire 
rope transmission has been shown. The various applications of 
the methods alread}' discussed exhibit much ingenuity and abil- 
ity on the part of the designers. At the same time there ap- 
pears to be a possibility of improvement, especially in the case 
of the transmission of large powers over long distances involv- 
ing a number of stretches. 

The Ziegler system of intermediate pulleys has given excel- 
lent results, but the following points may be enumerated as ob- 
jections ; 

a. The great height of the supports usually necessary because 
of the large size of the pulleys. 

b. The large base required for the supports, not only for clear- 
ance for the lower part of the rope, but also to resist the tension 
of the rope. 

r. The necessity of making the supports of great strength 
' when gearing is to be carried. 

These three points are all well shown in the Zurich station, 

FifT- 905 • 

d. The resistance due to stiffness of the rope. This has 
usually been considered uniinportant, uutil the recent investi- 
gations have shown otherwise. (See the preceding section.) 

e. The loss of power when the rope becomes slack. 

/. The necessity of giving sufficient tension to the rope to in- 
sure satisfactory action in warm weather and consequent exces- 
sive tension in winter. 

£■. The unsightly soiling of the exterior of buildings caused 
by the grease from the cable defacing the wall upon which the 
receiving pulley is placed. 

/;. The necessity of making the intermediate pulleys strong 
enough to carry the heavy stress of the cable, thus increasing 
the weight and consequently the journal friction. 

It therefore appears advisable to devise a system which should 
permit the supports to be made low and light, to use a light 
cable under moderate tension, also to reduce the number of 
splices, and to place the terminal pulleys inside of the building, 
the pulleys being made as light as practicable. 

All these points have been attained to a great extent in the 
following system. 

In the lirst p?ace, the cable, whenever possible, is made in 
one endless length from the driving to the driven pulley, thus 
making the intermediate pulleys merely supports and permit- 
ting them to be constructed very light. It is also desirable to 
arrange the cable so that both parts shall be at the same height 
from the ground and that this height should be as moderate as 
possible. 

In Fig. 90S is shown the arrangement of the power house, the 
first driving pulley 7", being directly upon thf motor shaft aud 
lying in a horizontal plane. The driving part of the rope then 
passes around a sta ionary pulley Zj and is c irried off in the 
desired direction. The driven part of the rop passes around a 
pulley /,' mounted on a carriage running on • track parallel to 
the direction of the line of transmission • ud by means of 
weights a pull somewhat greater than 2/ is rought upon the 
carriage. This tightener pulley /-' is placed >o as to bring the 
driven part of the rope to the same height as the driving part. 
The whole arrangement may be protected under roof as shown 
and the rest of the building used for other purposes, but if 
necessary the track and carriage maj' extend out of doors. 



The intermediate stations may all be supporting stations 
meiely, unless power is to be taken off at an intermediate point. 
If the transmission is a normal one, not using the method of in- 
creased tension (see g 293) the same deflection will be obtained 




in both portions of the rope by making the stretches for the 
driven part half as long as those of the driving part, so that 
every other station may be provided with a double-grooved 
pulley, Fig. 909. 



"X-lji Ln' ^10! 1-9! 

Fig. 909. 

If no change in direction is necessary the cable is thus carried 
to the driven pulley, the two parts being separated by a distance 
equal to the diameter of tbe driving pulley 7",, and entering the 
building where the power is to be received the cable passes over 
guide pulleys Z,.,, L-, and around the driven pulley T.,. 

When the load is reduced by throwing off machinery in the 
manufactory, the ti,ghtener carriage is drawn toward the turbine 
(Fig. goS) by the driving part of the rope, since both parts give 
a pull of j4 ( T-+ i). A spring buffer is provided to check the 
motion of the carriage in that direction. A spring dynamometer 
may be connected with the bearing of the other pulley Zj and 
the tension thus measured experimentally. When the trans- 
mission is set in motion 
from a state of rest the 
tightener pulley L moves 
slowly back until the 
tension in the driven 
part of the rope becomes 
equal to t. Should the 
rope have much stretch, 
the carriage must have 
sufficient travel pro- 
vided, and when neces- 
sary the rope must be 
shortened. The stretch 
of the cable is less ju this 
arrangement than with 
intermediate driving 
pulleys, because it is 
bent less frequently 
around the pulleys, and 
the wear of the rope is 
much reduced for the 
same reason. 

If angle stations are 
needed the arrangement 
of Fig. 910 is used ; this 
requiring only two pulleys to each part of rope, instead of three, 
as formerly, and the use of gear wheels is avoided. 

If the first driving pulley is in a vertical instead of a horizontal 
plane, the arrangement shown in Fig. 911 « is used, this requiring 
one more guide pulley than before. In this case the track for 
the tightener carriage is inclined so that its weight is used to 
produce the required tension. If it is desired to place the 
tightenerpulley horizontal the arrangement shown in Fig. 91 1 5 is 
used. In the cable of the Brooklyn bridge the tightener car- 
riage is provided with a brake in order to check the suddenness 
of motion due to variations of load. A friction device similar 




THE CONSTRUCTOR. 



207 



to o Fig. 709 will serve for this purpose if the angle 6 is made 
somewhat greater than is given by formula (233). 

If it is desired to place the driven pulley T^ in the same plane 
as one of the parts of the main line cable, the other part must 




Fig. 911. 

be led over another angle pulley. If power is to be taken off at 
intermediate stations these maybe constructed as the angle sta- 
tions of Fig. 910. 

Various other forms of intermediate power stations may be 
used without involving the use of gearing, as shown in Fig. 912, 




Fig. 912. 

in which a is for a shaft at right angles to the cable, and b and 
c for inclined shafts for either direction of revolution. 

The very moderate force which this system brings upon the 
supporting pulleys permits them to be made very light. This 
has been difficult of accomplishment with a cast iron rim. A 
light wheel can be made of wrought iron, using angle iron 
riveted to a special shaped centre piece, as shown in Fig. 914. 



Fig. 913. 



Fig. 914. 




These rims are bent hy means of special rolls, and a tongue is 
formed in the sides of the groove to hold the leather filling in 
place. The arms are made of light flat iron and the hub of cast 
iron ; the arms either being bolted fast or cast into the hub, the 



latter being made in halves. Pulleys made in this manner are 
very light. 

The construction of the supports is also peculiar, as shown in 
Fig. 913. The two posts are made of channel iron secured to a 
block of stone iu the ground by means of lead run in around 
the holes in the stone. The whole is steadied by guy-rods, and 
brackets are provided so that the bearings can be reached by a 
ladder. In many cases these supports of iron are cheaper than 
those built of stone. 




Fig. 915. 

For the intermediate driving pulle5-s of cast iron, the form 
shown iu Fig. 915 is used. The hub is outside of both bearings, 
but the plane of the pulley is midway between the journals. 
The connection between the arms and the hub is made by 
means of a hemispherical shaped device, somewhat resembling 
the frame of an umbrella, and hence these have been called 
" umbrella " pulleys. This construction enables the pulley to 
be firmly secured and readily removed without disturbing either 
bearing. 

In Fig. 91 5 b, a modification of this form of pullej', the 
umbrella-shaped hub being made separately, and a straight arm 
pulley fitted upon it. This permits a single pattern to be used 
for the centres of a number of sizes of pulleys, or wrought iron 
pulleys may be used on cast iron hubs of this form. Instead of 
two journals a single longer one may be used, two forms of 
hangers being shown in dotted lines. 

The use of the umbrella 
pulley enables a verj- sim- 
ple form of support to be 
used, either for single or 
double stations. 

Fig. 916 «, is a single 
station composed of a 
wooden post upon which 
a projecting bearing is 
bolted, and iu which the 
journal of the pulley runs. 
At b, is a double station, 
the post being made of 
iron The dotted lines at 
D indicate a small roof 
to protect the bearings 
from the weather. A 
comparison of these forms 
with the older style, as 
for example. Fig. 903, will 
show that merely the use 
of the continuous rope 
and the umbrella pulley 
will effect a great econ- 
omy in construction. The 
umbrella pulley is also 
well adapted to be used 
for rope sheaves for hoist- 
ing machinery and for 
chain sheaves.* 




Fig. 916. 



* Various applications of the umbrella pulley will be shown hereafter. 
The principle is also applicable to bell pulleys. At a, is a simple counter- 



208 



THE CONSTRUCTOR. 



300 will be a practical 



A coraparative example with that in 
illustration. 

Exa^nple. — The transmission at Obernrsel is made in eight equal stretches 
and seven stations with two pulleys each, one driving pulley and one driven. 
This gives 16 semi-circular wraps of the rope about the pulleys, causing a 
loss of 5.13 H. P. from stiffness. Bj' the adoption of the new system there 
would be three semicircular wraps at the power house (see Fig. 908), one on 
the driven pulley and two quarter.wraps ou the guide pulleys L^, /.y (see 
Fig. 909) There are also 11 short arcs of con tact, about 3V o^a circle each, on the 
supporting pulleys, which latter would be very light and on supports con- 
structed as already described. The combined arcs of contact make practically 
about 5 semi-circular wraps or /g of the resistance of the old arrangement, that 
isTe- 5. 13 or about 1.6 H. P. This is not too I'avorable an estimate, as we have not 
included the effect of the excessive tension which often occurs by the con- 
traction of the cable in cold weather, and which is entirely avoided by the 
use of the tightener pulley and carriage. 

The reduction of journal friction is also important, as the weight of the 
puUevsand the effect of the rope tension are both much less. The total 
weight of the pulleys will be only about 7^ that of the old system, although 
more pulleys would be used, and the journal diameter may be reduced to J^ 
of the previous value. This gives a loss of 3 X % = g of the previous value of 
9.36 H. P., which is2.oS H. P. To this we must add a resistance of 0.40 H. P. for 
the guide pulleys which have been added in the new system, giving a total loss 
ofi\^2s =1.60 4- 2.0S 4-0.40= 4.08 H. P. The loss in the first instance is with the 
new system 4 per cent, and in the second 10 per cent., as against 13 9 and 35,9 
per cent, for the old system. 

In this example there are no intermediate power stations, the 
entire amount of power less only the hurtful resistance. In 
considering the question of the stress in the driving part of the 
cable it is important to know whether the entire power is to be 
transmitted to the etid of the line or if a portion is to be taken 
off at intermediate stations. If the initial forces at successive 
intermediate power stations be indicated by /",, f,,, P^,, -f^, etc., 
the successive tensions in the cable will be reduced, and hence 
the deflection h should be determined for the stretches preced- 
ing and following each station, and the tension in the cable will 
vary according to the power taken off at intermediate points. 
The sum of all the forces P, will in every case be determined by 
taking the tension t, in the driven part at the first driven pulley, 
from the initial tension T, so that we have T — t ^=^ S P. From 
this equation we can deduce important results. 

As an illustration we can assume the entire power transmitted 
to be divided up among a number of intermediate stations, all 
being operated by one continuous cable, as shown in diagram 
in Fig. 917. 




Fig. 917. 

In this case the rope passes the entire round of stations T^i, 
7",, T3, T^ to Tji, returning to the main power house. The 
rope returns to the power house at any angle with a tension I, 
giving T^ J, P -\- i. All stresses are regulated automatically 
for each stretch of the rope, as the forces vary at each station. 
If the work at any station is reduced or even becomes zero, the 
tightener carriage responds and alters the deflection so that 
T — t ^ "2. . P, in which i remains constant. A transmission of 
this kind, in which the cable makes a complete circuit of a num- 
ber of stations, maybe called a "ring" system. In Fig. 917, the 
supporting stations are indicated by small rectangles or tri- 
angles, according as the line is straight or makes an angle, and 




shaft, at (5, a simple headstock for a small lathe, and at c, is a head for a 
boring machine, the loose pulley running on a stationary sleeve, as already 
shown i;i Fig- 862. 



the power stations as shown are circles. At 7"g the rope passes 
off into an auxiliary circuit, which may be called a "ring" 
transmission of the second order (see § 260). The stations may 
all be constructed very simply. The supporting stations are 
made with one pulley when the line is straight, and with two 
at the angle stations ; the power stations can generally be made 
with only two pulleys, providing the necessary arc of contact a, 
is obtained, or three pulleys used if necessary, see Fig. 918. 




Fig. 91S. 

In many cases it is desirable to use the system for under- 
ground transmission, as in Fig. 919.* 




Fig. 919. 

In order to determine when an arc of contact a , of the 
proper magnitude has been obtained, we have, from (239), \{ P 
is the greatest force to be transmitted by the pulley with a ten- 
sion T' : 



P-- 



We will 



P 



T' 



call the ratio ^, which is the reciprocal of the modu- 
lus of stress, the modulus of transmission, and let it be repre- 
sented by 6, whence : 



ef'<^ 



if/'a 



(309) 



Neglecting the influence of centrifugal force, we have, from 
g 290, for f the values f ^= 0.22 and 0.25 to consider. Taking 
these we get the following values for various angles : 









Modulus 


OF 


TRANSMISSION 


e. 








0( = 


15° 


30° 


45° 


60° 


90° 


120° 


150° 


180° 


270° 


360° 


450° 


540° 


_/= 0.22 
= 0,25 


0.06 
0.07 


O.II 

0.12 


0.16 
o.iS 


0.21 
0.24 


0.29 

0-33 


0.3S 
0.41 


0.44 
0.48 


0.50 

0-54 


0.65 
0.69 


0-75 
0.79 


0.86 
0.81 


0.88 
0.87 



These values are shown graphically in the following diagram, 
Fig. 920 : 











2. 





360 


460 


5-t 


a 




































~ 
















, 




Ji^ 










U,b 




















,.i^ 


-r^ 


.2'.* 












J 


1 






( 






, 


^n^'"- 


^^^■^ 










- ' 




















..^ 


r^ 


t!S5» 








' 






















K^ 


^ 
































\^ 






















> f 








/' 


r 


























y 






/ 






























Q. 


/ 


^ 


1 
1 



























Fig. 920. 

From this it will be seen that an arc of contact of 30° will per- 
mit the transmission of jij the power due to the tension T\ and 
an arc of 90° gives about 5-3. 

A convenient application of this principle is found in the 
arrangement of a " ring" transmission when a large arc of con- 
tact is obtained upon the first or main driving pulley by redu- 



* This has been done in San Francisco by Boone, using a conduit for the 
rope similar to a cable railway. 



THE CONSTRUCTOR. 



20Q 



plication of the rope over a counter pulley, as in Fig. 795, and 
also shown in the case of the double-acting belt transmission in 
Fig. S60. B3' using a single-grooved counter pulley and double- 
grooved driver we get ot i 360°, so that is at least equal to 0.75. 
In this way the specific capacity of the rope can be materially 

increased, practically about iji times. If we give r = -— the 

value -|- in the first equation of \ 290, we have for the specific 
capacity of a cable transmission with a counter pulley : 



iV„ = - 



■s-i 



or say N^ -- 



33000 
25000 



24750 



s„ 



(310) 



The adaptatiou of the mechanism to receive the counter pulley 
is usually not difficult. 

The adaptability of the "ring" system of transmission to use 
in- distributing power in manufacturing establishments is appar- 
ent, and for this purpose hemp rope is very suitable. This will 
be shown by the following example : 

Example r.— The transmission shown in Fig. 8Si, §286, contained 16 hemp 
ropes 2 inches in diameter, each having- a specific capacity Nq ~ 0.0021 and 
11 = 2360 feet per minute. The cross section of each rope is 3.14 sq. ins. 
Hence iV = Nog v = 0.0021 )< 3.14 X 2360 = i5-57 H. P. for each rope, or 
2^9 H. P. for the 16 ropes. 




Fig. 921. 

Substitutinpf the arrangement shown in Fig. 921, we take a single wire 
cable composed of 60 steel wires, and use a stress of 17,000 pounds in the 
driving side of the cable and increase the speed to 3150 feet per minute. We 
then have from (278) : 



: 66,000 - 



249 



17,000 X 3150 
and hence the area of each wire is 



= 0.307 sq. in. 



60 



- 0.0051 sq. in., and the diameter 6 = 0.08", 



In the original hemp rope transmission the main driving pulley had a 
radius of 71 inches, and as we have increased the speed ^ times, the driving 
pulley must be proportionally increased, and hence the radius will be 95". 
This gives a stress due to bending, 



12,000 lbs. nearly; see formula (279), 



;9s 



this being not too great to 'give satisfactory results. We have, instead of a 
wide face puUev made with 16 grooves, a single groove pulley made with 
leather filling, as in Fig. S79 a, of 15 ft. 10" diameter. An important point 
to be considered is the stress due to the bending of the rope over the pul- 
leys Ti, T2, etc. These pulleys were 36" radius for the hemp rope, and 
hence *. 36 = 48" radius for the wire rope, or 8 feet diameter. We then have 
from {279) 



0.08 
"76~ 



23,666, which added to the working stress of 



17,000 lbs. gives a totaFof 40,666 pounds, which is not too high for steel wire, 
according to § 266. The idler pulley L, is made the same size as the driven 
pulleys T'g. T^, etc., and the tightened pulley Z.' can be made a little larger. 
The loss of efficiency will be somewhat less than in the case of hemp rope, 
since for wire rope there is a smaller modulus of stress t, (/. e. 2 instead of 
25, see 'i 287), and the initial force P, is smaller, because of the increase in 
velocity and the loss from stiffness will be less. The loss from stoppage 
and creep should also be considered as not unimportant (see § 287). 




^^^^^^^^^^^^a 



Fig. 92 



If it is desired to use a counter pulley with theabove transmission, the ar- 
rangement in Fig. 922 may be adopted. In this case the counter pulley G, 
and tightened pulley /.', are both inclined so that the rope shall be properly 
guided for the double grooves m the main driving pulley. The arc contact 
a is in this case greater than 360°, and the specific capacity will be ij^ times 
greater. This will enable the cross section of the rope to be reduced to 3 the 
previous value, or ^ = ^. 307 ^ 0.204 sq in. If we use wires instead of 60^ 
we have for the cross section of each wire 



0.204 

36 



■ = 0.005*^ sq. in., and 6 = 0.084 i^* 



The diameter of the rope will be from 8 to 9 5 or %" to £", the latter when 
the rope is new. 

The conditions of this example are hardly such as to demand! 
the introduction of the counter pulley, but when large powers, 
are to be transmitted its use is most advantageous. In some 
instances the counter pulley may be arranged, as in Fig. 911, 
so as to sustain a part of the weight of the fly wheel of the en- 
gine, and hence materially reduce the journal friction. 

In many instances the power in factories may be arranged so 
as to use the "ring " system of transmission, and dispense with 
the use of the spur or bevel gearing, and some examples are 
here given. 

In Fig. 923 a is shown the usual arrangement of the trans- 
mission of power in a weaving establishment. 



■-dl- 



-^ 1 



Fig. 923 a. 



In this instance the two shafts which extend each way from 
A', drive the line shafting by seven pairs of bevel gears, while 
in some factories as many as 12 to iS pairs are used. 




Pig. 923 b. 

Fig- 9-3 b shows how a ring transmission can be used to drive 
the same shafting, there being seven guide pulleys and one 
tightener L', the guide pulleys being of the "umbrella " pat- 
tern, as in Fig. 915. The tension weight for the tightener is 
equal to 2 T'. , 




Fig. 923 c. 

Another arrangement is shown in Fig. 923 <-, this being used 
when the alternate shafts are to revolve in opposite directions. 
This permits the rope to be used double acting, as described irt 
? 277 and shown in Fig. S60. Those portions of the rope 
marked i in Fig. 923 c, are in one plane, and those marked 2 
in a second plane, giving clearance to the parts of the rope, 
and the rope is guided from one plane to "the other by the 
guide pulley Zj and tightener L'. Five of the seven driven 
pulleys are double acting, and hence are made double grooved. 



2IO 



777^ CONSTRUCTOR. 



Shafts which lie at right angles but in parallel planes, one 
above the other, are also readily' driven by use of a ring trans- 
mission system. 





Fig. 924. Fig. 925. 

In the preceding cases it is desired to obtain a double wrap 
of the rope about the driving pulley A', the arrangement in 
Fig. 924 may be adopted. In this case two idler pulleys G-^ and 
<?, are used to guide the rope from one plane to the other. The 
rest of the rope, when either of the planes shown in Fig. 923 b 
or c is used, is guided in a third plane by suitable pulleys. In 
Fig. 925 is shown an arrangement by means of which a series 
of parallel vertical shafts, revolving alternately in opposite di- 
rections, can be driven from a single horizontal shaft K. 




Fig. 926. 

The ring sj^stem is well adapted for driving a number of mill 
stones, as arranged in Fig. 926, for example, in which all the 
mill spindles revolve in the same direction. The direction of 
the stones ma}' be readily reversed by a corresponding change 
in the cutting of the furrows, and hence the double-acting 
arrangement as in Fig. 925 can be used if so desired. 

The arrangement of the double-grooved pulley on the spin- 
dle in this case is shown in Fig. 927. This is a modified form 
of the umbrella pulley of Fig. 915, the hub being made in the 
form of a hollow sleeve carrying a cone or other suitable clutch 
K, K' , by which any pair of stones can be stopped without in- 
terfering with the 
others. An adjust- 
able step for the 
spindle is provided 

2XH. 

In machine shop 
transmissions it is 
frequently required 
to drive a series of 
parallel counter 
shafts, which re- 
volve in one direc- 
tion, sometimes in 
the opposite, and 
any of which may 
need to be stopped. 
Such an arrange- 
ment is shown in 
Fig. 92S. The rope 
is carried over the 
various puUej-s of 
one series around 
the tightener pulley 
/.', and back over 
the other series At 
/v", and A'2 are fric- 
tion clutches which 
are thrown into en- 
gagement on one 
side or the other, 
according to the di- 
rection of revolution 
required, or which may be left disengaged. If two adjacent 
shafts are desired to revolve in the same direction, an inter- 




FiG. 927. 




Fig. 92S. 

mediate guide pulley is introduced, as shown at L^. The sub- 
sequent belt transmission from these counter shafts can be 



greatly simplified by using the above sj-stem. In all these ar- 
rangements the modulus of transmission is determined as al- 
ready discussed in formula (309) and the proper arc of contact 
determined. For example, if in an arrangement similar to 
Fig. 925, each part of the rope is in contact 30° on the pulley, 
and the coefficient of friction f, is 0.22, we have from the pre- 
ceding table for the modulus of transmission ft = o.ii. If the 
tension on the respective sides of cable be T' and V , acting 
upon the puller's, we have for the maxinmm force transmitted 
by the rope P' ^o.w (T' -\- i'). In this case we have always 
jy J^t= T+ t, and hence T=2 2 P^t^^LP (see § 264). 
Hence we have P' = o.ii X 3 S P, or about -|- S P. If there 
were but three driven pulleys, each offering the same resist- 
ance, the system would operate well, and still belter with a 
greater number of driven pulleys. For mills of 20, 30 or more 
pairs of stones, this arrangement is especially applicable, since 
it furnishes a far simpler transmission system than heretofore. 
This system, however, should not be carried beyond its proper 
limits, and for small, light running mills, such as are used for 
grinding paints, graphite, etc., belts are generally more advan- 
tageous, being easier thrown in and out ; the rope system 
being better adapted for the transmission of greater powers. 

In all the various classes of heavier mills, such as are used 
for grinding plaster, cement, and the like, also for paper mill 
machinery, the rope transmission is best adapted, replacing all 
the heavy shafting, gearing and belting otherwise necessary. 

An example will illustrate the method of applying the fore- 
going principles. 

Example 2. — Let there be two sets of wood pulp mills each requiring 60 
K. P. to be driven from a pair of turbines by a "ring transmission " system, 
the first moving shaft making 125 revolutions per minute. The main shaft 
is driven by the two turbines by means of spur gearing, and carries the 
driving pulley of the rope system, making also 125 revolutions. "We have 

for the specific capacity of the rope, from (277) N^ = ^ , and ifwe use steel 

wire and take Si = 21,300 we get A'q = 0.323. .A.lso we make the velocity v of 
the rope = 3150 leet per minute, and we get for ihe cross sectiou of the rope 
from (278) 

N 120 „ . 

g = — — = — = o-iiS sq. in. 



vjV^ 



3150 X 0.323 



If we make the cable of 36 wires we have for the cross section of one wire 
O.I 1 8" , , 
; — = 0.0033 and the diameter S — 0.073 . 

From the number of revolutions and the chosen speed of rope we have for 



the pulley radius, J? = 
bending stress, 



3150 ■< 12 



: 4S'', and using this in (279) we get for the 



■ 14,200,000 — = 21,500 lbs., which is satisfactory. 



The 



entire 120 H. P. is carried by the rope to the first set, where 60 H. P. is used, 
and the balance is transmitted, to the second set, the necessary supporting 
pulleys being introduced between the two points, and the required tension 
/ being given bythe tightener carriage. Following the course of the rope, 
we have at the driving pulley the tensions 7" and /, respectively equal to 
2 y, Pand 2 -P, whence S /== ^j°°° X 12 ■ ^^^^ j^^^^ ^ ^ ^^^ j.^ ^^^^ X 2 = 



3'5o 

2514 lbs. The tension at the first set is reduced by P' = 628.5 lbs., whence 
T' = 2514 — 628.5 = 1S85.5 lbs. At the second point again is taken oflT P' = 
628.5 lbs., and the tension becomes 18S5.5 — 628.5 = -lz^-j lbs., which is equal 
to the above value of /, and which is obtained by loading the tightener car- 
riage with 2514 lbs. or a little more. 

This system requires the use of clutches for starting and stopping the 
machines, and for this purpose Adj-man's coupling, (§ 307), is suitable. 

In some instances it is found practicable to drive two pulp 
mills with one pulley, the pulley being between the machines 
on an intermediate shaft with a fraction coupling at each end. 

Another case may be given where a number of machines with 
horizontal shafts each requiring the same amount of power, -are 
arranged in a row aud drawn by a ring transmission system. 
Fig. 929. 




Fig. 929. 

In this case friction clutches are placed at A'j K^ for stopping 
and starting the machines, while the intermediate pulle3'S L L, 
which may be of the umbrella pattern, are carried on hangers 
from the ceiling. The rope and the driving pulleys are covered 
by guards S S to protect the workmen. This arrangement is 



THE CONSTRUCTOR. 



2ir 



especially convenient if there is a second series of machines on 
the floor above, when the pulleys L L become the driving 
pulleys of the upper set, and no guide pulleys are required at 
all. It is sometimes desirable to make the driving pulleys of 
umbrella form, supported on independent bearings, so that any 
machine can be repaired or entirely removed without inter- 
fering with the rest of the transmission. 

It should not be forgotten that the ring system of rope trans- 
mission generall}' involves an entire rearrangement of the 
establishment, and that it can rarely be substituted for a shaft- 
ing transmission to much advantage. 

A comparison of the last example with the older system in 
which a separate rope is used for each portion of the transmis- 
sion will be of interest. In the previous method the pulle3-s 
could not be brought close together because the tension would 
require to be too great, and slight variations in temperature 
would produce excessive variations in tension. These difficul- 
ties are overcome in the ring S5'stem by the use of the tightener 
carriage, which may also be used to much advantage in those 
systems of belt transmission which lie in one plane, such as 
have been shown in Fig. S44. The construction is similar to 
that for rope transmission, and the umbrella hub may be used 
to advantage. In many cases the specific capacity may be much 
increased in this way. 

The new system is also highly advantageous for long distance 
transmission, especially where power is to be taken off at 
several points, or it may be used in combination with the old 
system, retaining the latter and using the new system for pur- 
pose of distribution. 

The difficulties of construction are much less for long dis- 
tance transmission than with the old system, and the cost of 
installation and supervision much smaller. 

The application of the new system appears likely to increase 
very greatly, since it involves less first cost than electrical 
transmission plant, and also a higher efficiency when the losses 
from transformation of electrical currents are considered. 

This subject will be further considered in Chapter XXIV. 



CHAPTER XXII. 

CHAIN TRANSMISSION. STRAP BRAKES. 



mation may be obtained by taking /3 =^ — , which gives for the 
modulus of friction p : 

In chain transmission the modulus of friction is not indepen- 
dent of r, as with rope transmission, but varies somewhat with 

the ratio — . This latter ratio in practice seldom goes below 5. 

Taking this limit, and also puttingy = o.i, we have for practi- 
cal values of p the following, in which « equals the number of 
half wraps of the chain around the sheave : 



■ = T=0 + i 



-s 5 "■ « 



whence : 



T 



= 1-37" • ■ 



(3-3) 



The following table has been calculated for i to S half-wraps, 



and gives the modulus of friction p ■■ 



T 



, the modulus of stress 



T 
r^ — and the modulus of tiansmission (see (309)). 



u = 


I 


2 


3 


4 


5 


6 


7 


8 


p = 


1-37 


1.8S 


2.57 


3-53 


4-83 


6.6i 


9.06 


12.41 


r = 


3-69 


2.13 


1.64 


1-39 


1.26 


1. 18 


1. 12 


1.09 


e = 


0.27 


0.47 


0.61 


0.72 


0.79 


0.S5 


0.S9 


0.92 



Specific Cap.\city of Driving Ch.\ins. 

The use of chain for purposes of power transmission is neces- 
sarily more restricted than the use of rope, but for single trans- 
missions in special cases it is well adapted, and its applications 
are increasing. Chain is especially capable of resisting varia- 
tions of temperature and exposure to the weather and to dust, 
and hence is well adapted for driving revolving drums in 
mining machinerjf, washing machinery, the machines in bake- 
ries, etc. In mining machinery chains are very extensively 
iised, both above and below ground, not only for continuous 
tramway driving, as in Fig. S02, but also for the transmission of 
rotary motion over extended distances. 

Chain sheaves are made either with smooth wedge shaped 
grooves, or with pockets for the chain links as already indicated 
in J 275. In the first case the driving is due to friction in the same 
manner as with belting and ropes, while in the second case the 
action is similar to that of toothed gearing. 

a b 

I 





Fig. 930. 

The method of friction driving can be used with ordinary 
link chain as at n, Fig. 930, and may also be used with the fla't 
link chain of Fig. 830 d, if so desired. The circumferential 
friction F ■=^ T — t may be determined from the following rela- 
tion (see \ 264) : 



T 



— ( 



(i -|- 2_/sin 



4)' 



(311) 



in which 7^ and t are the tensions in the driving and driven 
sides of the chain respectively, and / is the coefficient of fric- 
tion. The angle /i is that subtended by the pitch length of a 
link of the chain at the centre of the chain sheave, and may be 
obtained from r sin yi j3 = yi ! ; the exponent iii is the number 



of link contacts, hence Jii = 



/3 



A sufficiently close approxi- 



These values for p and r are similar to those obtained for ten- 
sion organs generally, as indicated in the diagram already given 
in Fig. 816. It will be noted that the transmitting capacity of 
chain even with a single half-wrap about a smooth sheave is 
good. 

Since the specific capacity of a driving tension organ (see 
(262)) is equal to 

jv^ :=— — . — or — — se 

33000 r 33000 

we have for ordinary open link chains the following values for 
various stresses 5: 



5000 
7000 
8500 



« = 


I 


2 


3 


4 


5 

0.120 
0.168 
0.203 


6 




0,042 
0.057 
0.070 


0.071 
0.099 
0.121 


0.092 
0.129 
0-157 


0.109 

0.153 
0.185 


0.129 

0.180 
0.219 



0.135 
0.189 
0.230 



0.140 

0.197 
0.237 



The specific capacity is in all cases high, and for the generally 
accepted stresses in the chain cross section it varies from 
0.042 to 0.237. Various applications permit variations in 
the value of S, the value being taken lower when it is desired 
that the wear through friction shall be reduced The cross sec- 
tion of chain is determined from the equation N =2q v N ^ 
(see § 2S0) in which N is the horse power to be transmitted at 
a velocity V, and q is the sectional area of the iron of which the 
chain links are made. We have : 



? = : 






(314) 



The value of v is always low, and hence the influence of cen- 
trifugal force upon /'j may be neglected. 

Example i. — It is required to transmit lo H. P., by means of a chaiu 
making a half wrap about a smooth sheave, the velocity z' being iiSo feet per 
minute and S = 8500 lbs. We then have for the cross section q of metal : 

= 0.0609 sq. in. 



2 X 1180 ' 0.070 
which corresponds to a diameter of 0.3 in. 



212 



THE CONSTRUCTOR. 



Example 2. — By using the counter sheave (Fig. 795) and thus obtaining 
three half-wraps the value of 6 may be reduced to 5000 lbs., whence 



? = ; 



: o 046 sq. in. 



2 X 1180 ' 0.092 
or a diameter of 0.27 in. 

This gives a lighter chain and at the same time a more durable one, as the 
friction is materially reduced when entering and leaving the sheave (see 
§ 303)- 

By using grooved and pocketed sheaves the specific capacity 
may be greatly increased, the chain being held so securely that 




_.^g^ 



Fig. 931. 

as many as eight half-wraps may be used. Two very practical 
arrangements for such sheaves are shown in illustrations, which 



are from executed examples in the chain tramway of the Decido 
iron mines in Spain, built by Briill, of Paris. The dimensions 
are given in millimetres, and the chain is operated under a 
stress of about 5000 pounds per sq. in. 




k_.20J}....>; 



Fig. 932. 

The sheave shown in Fig. 931 is for a 25 mm. {\" nearly) 
chain, and is made with inserted teeth of steel, and the form of 
Fig. 932 is similar, and is for an iS mm. (0.7 in.) chain. In both 
cases the teeth are radial, and formed to recjive the chain links, 
being secured by jam nuts in the second case, and by nuts 
fitted with the Belleville elastic washers, which latter have 
worked well in practice. 




Fig. 933- 



THE CONSTRUCTOR. 



213 



In Fig. 933 is given an arrangement of chain sheave gearing, 
including a solid massive form of bearing, as used in many 
English collieries.* Here the sheave is made with eight semi- 
circular ridges or ribs, similar to the old form of capstan shown 
already in Fig. 794 a ; and both parts of the chain are carried 
on supporting pulleys. In many instances this arrangement is 
used, by widening the face of the sheave, to receive several 
wraps of chain, as shown in the upper right corner of Fig. 933. 
If we may safely assume that the ridges increase the coefficient 
of friction at least three times, in the preceding formulas (311) 
and (312), we have for the corresponding modulus of friction p' : 



which gives for 



p' = 2.5!' 



(315) 



In this case the pitch length / of the links is taken = 3.5 d, 
and making r = 5 ?, we get ^1 = ( 7"+ o-OS^yi, and if we put 
for the loss at both sheaves : 



P 



= Ek 



we get : 



Ek = 0,072 /i 



P+i 



(316) 



Example t. — Taking the coefficient of friction ^^ ~ o-^5 on account of the 
small bearing surface we have for a chain transmission on smooth sheaves 
with half-wrap ; p being = 1.37, as in the preceding section : 



6' = 



}i 


I 


2 


3 


4 


1.58 


2.50 


6.25 


15-63 


39.06 


2.72 


1.67 


1. 19 


1.07 


1.03 


0-37 


o.5o 


0.84 


0.94 


0.97 



£■4 = 



from which the security against slippage and also the specific 
transmitting capacitj' may be determined for any given case. 

Within moderate limits chain transmission may be used as a 
" ring " system, as for instance in driving the rollers of carding 
machines, also in wood pulp grinding mills a ring chain trans- 
mission is used for driving the feed rolls. 



Efficiency of Ch.^in Transmission. 

The loss of efficiency in a chain transmission is due to jour- 
nal friction, dependent upon the chain tensions 7"and f ; and 
upon the friction of the links in entering and leaving the 
sheaves. The journal friction is determined as already shown 
in § 300, and for high values off?, it is not great. The loss from 
chain friction is due to the rotation of each link about its 
adjoining link as an axis through an angle j3. This gives, with 
a coefficient of friction /i, a circumferential resisting force /^„ 
due to chain friction (see formula loo) 



/^i=/, T+ t 



© (4-)- 



2.37 

: 0.072 X 0.15 = 0.0692 

or say 7 per cent. 

Example 2. — If the sheave is made with ridges, as in Fig. 933, we have \.i 
= 2.5, and hence 



Rk = 0.072 X 0-15 



3-5 
1-5 



= 0.025 



* The illustration is from Newcliurch colliery at Burnley. 



or only 2)^2 P^r cent. 

Example 3. — By using carefully :nade pocketed teeth and making n = 8, 
we have p = 12.41, whence 

Ek = 0.072 X 0.25 ( — - — 1 = 0.0126 
\ii.4i/ 

or only i^ per cent., this reduction being due to the reduction in the tension 
on the chain, showing the importance of considering the question of chain 
tension in this connection. 

In the preceding e.'^amples the friction of the links upon each 
other has been considered, but not that of the links upon the 
sheave. This latter is a very variable quantity, being unimpor- 
tant with a smooth sheave, as Fig. 930 n, and sometimes 
becoming excessive, as shown already in Fig. S3S b, \ 275. 

In every case all possible care should be taken to produce as 
little rubbing contact as possible. 

I 304. 

INTBRMEDIATB STATIONS FOR CHAIN TR.4.NSMISSI0N. 

The most important applications of chain transmission are in 
mining work, both above and below ground ; and especially in 
coal mines. In this branch of work England takes the lead, 
followed by America, where, however, wire rope is more exten- 
sively applied, while in Germany the most applications are 
found in the Siiarbruck district. 

A very interesting application of endless long distance chain 
transmission is shown in Fig. 934, which gives two views of the 




^fw.ii'/ '///M/////y', 



Fig. 934. 

(Dimensions in Metres.) 



214 



THE CONSTRUCTOR. 



Ganuow mine at Burnley in Lancashire. The driving pulley 
is at T, and guide pulleys at L, while at L' is a tightener pulley 
iung between two idlers, a construction which is frequently used. 
The rotation is modified in various ways in the English mines, 
stations similar to those of rope transmission systems being used. 




Fig. 935. 

In Fig. 935 is shown an intermediate station at 7", T.,, and 
also on angle station at Z. In manj' instances combinations of 
bevel glaring aud shafting are found in connection with chain 
transmission, but the examples here given are confined to the 
use of chain alone. 




Fig. 936. 

In Fig. 936 an intermediate station is shown at T^ T^, and a 
change station at T-^ T^. At 7"j, Fig. 936, the chain makes an 
entire wrap around the sheave, the latter being made with a 
wide groove, and interference of the two parts of the chain pre- 
vented by guide sheaves. The simple supporting stations are 
made with small horizontal guide sheaves, with wide grooves. 
The velocity of the chain varies from 200 to 500 feet per minute. 

? 305. 
Strap Brakes. 

If a driven pulley is embraced by a tension organ, either belt, 
rope, strap or chain, the ends of which are subjected to tensions 
7" and /, and also held from moving, the pulley is hindered from 
moving toward /, so long as the force acting to rotate it does 
not exceed P=r T — t. The tension organ then forms, with the 
pulley and stationary frame work, a friction ratchet system in 
which the tension organ forms the pawl. If the tension 7" be 
reduced until T — t <^ P, the pulley will slip in the strap, over- 
coming the frictional resistance due to T — /, and the motion 
can be made slower, if 7" and t be made great enough, so long 
as their difference is only slightly smaller than P. The mechan- 
ism then becomes a form of checking ratchet (? 253) better 
tnown as a friction brake, or simply as a brake. Such brakes, 
when made with tension organs, are called strap brakes. 

Strap brakes are made in various forms to suit the applica- 
tion. 

"• b. 






Fig. 937. 

(«■) Clamping Brakes. — When a strap brake is to be used to 
act as a complete clamping brake, to check motion entirely, the 
tensions 7" and must be determined. These are obtained from 



formulas (239) and (240) or from the graphical diagram of Fig. 
816. Such strap brakes are frequently made with straps of iron 
or steel. It is generally desirable to so arrange the parts that 
the motion of the pulley acts to draw the strap into closer 
engagement, which may be done in various wa3'S. Fig. 937 
shows several such arrangements. 

The various parts are indicated as follows : i is the axis of the 
pulley ; 2, the point of application of the brake ; 3, the attach- 
ment of the tight side of the strap ; 4, the attachment for the 
slack side ; 5, the axis for the brake lever. In Fig. 937 fi, 3 and 
5 are separate ; in Fig. 937 b they are combined in one, and in 
F'ig. 937 c both 3 and 5 are separate, but 3 and 5 are made mova- 
ble, and 3 and 5 are so nearly in line with 7" that a very slight 
effect is produced on the lever by T. 

In Fig. 938^ 3 aud 4 are combined, aud at the same time 3 
and 5 are nearly in line with T. Fig. 93S b is the so-called. 






Fig. 938- 

"differential " brake of Napier, in which 3 and 4 are so placed 
that perpendiculars to the directions of T'and t are inversely 
proportional to those tensions, thus reducing the action of the 
strap upon the brake lever to a small amount. Fig. 93S c shows 
an arrangement adapted to permit the pulley to revolve in 
either direction. The angle 3.5.4 can be so chosen that the 
force upon the lever may be very small. 

For heavy hoisting machinery, the braking power required 




Fig. 939. 

makes the arrangement shown in 939 suitable. In this case the 
strap is filled with blocks of wood in order to obtain a higher 
coefficient of friction aud at 6 is shown an application of the 
globoid worm and worm wheel shown in Fig. 641. 

Example. — Required a brake for a shaft driven by a force of 2200 pounds at 
a lever arm of 7.875 inches. The form chosen is that of Fig. 938 a. the arc of 
contact a of the strap being 0.7 of the circumference. The coefficient of 
friction 7= o.J> the strap being lubricated. M'e then havey"c/ =0.1 X i 4 w. 
= 0.14 77 = 0.43. We then have from the second table of § 264, the tension 



modulus T - 



T 
P 



: 2.SS nearly, and the friction modulus p = — ~ = 1.5 (see 



t 7 

also the diagram. Fig. 816). This gives -75- = ^ . —^ ^ ?i X 2.88 = 1.92. If 

we make the brake pulley with a radius of 15.75 in., the braking force at the 

circumference of the pulley must be — — ^ . 2200 = iioo lbs., and t = 1.02 V 

15.75 
IIOO = 2112 pounds, and 7= 2.88 X 1100 = 3168 pounds. If the brake is to be 
operated by a hand lever with a force of 44 pounds, the ratio of the length of 

the hand lever to lever arm 4 . 5 must be .^^ =^ 48. The strap is under a 

44 
tension of T = 316S pounds. If we assume a permissible stress of 5 = 14,220 
lbs. and a thickness of strap 6 = o.oS" the width will be; 



316 



= =78", 



which is quite practicable. 



THE CONSTRUCTOR. 



215 



we have for the tight end, where 



The question of the pressure between the braking surfaces is of interest. 
P _ Q 
b' R 
S^ 14,220. 

o.oS 
p = 14,220 —^^rZT — 7^ ''^^* 



According to formula (241) -~ = 



15.75 



and at the slack end, since — - 



Fig. 940. 



■ ^, > = ^ . 72 =; 48 lbs., both of which 
such small values that the wear must be very slight. 

This example shows how, in a properly arrauged construction, 
a great ratio of force to resistance can be obtained. In large 
winding engines the brake pulley can readily be cast in one with 
the rim of the drum gear. 

The method of securing the ends a b 

of the metal strap is shown in Fig. 
940. The form at a, is secured by 
countersunk rivets, and that at 5, 
by an anchor head and a single 
small rivet to prevent lateral slip- 
page. 

(b) Sliding Brakes. — In using clamp brakes operated by hand 
for lowering heavy loads in hoisting machinery, great care must 
be taken, since the throwing out of the checking pawls puts the 
entire resistance on the brake. With this arrangement there is 
always more or less insecurity, the safety depending upon the 
handling of the lever, and serious accidents have frequently 
occurred. This danger can be avoided by the use of automatic 
sliding brakes, the following form being designed by the author, 
and shown in two forms in Fig. 941. The brake pulley a, is 
loose on the shaft, but engages with it by means of a ratchet 
system a' b' c' ■ The brake is subjected to a tension equal to 

a. b. c. 

JL_ 3. A^-'_a 




Fig. 941. 

the greatest braking force desired; i.e.s,o that the weight K 
must be raised in order to permit the load to run down. If the 
lever is let go, for any reason, the descent is checked. In form 
a, the pawls are attached to the pulley, and the ratchet wheel 
a' keyed to the shaft ; in b, the pawl is on a disk c' . When the 
load is raised the combination forms an ordinary ratchet train. 
A silent ratchet. Figs. 673, 674 may be used for this device. At 
c, is shown a pendulum counterweight, which can be adjusted 
so as to vary the braking power to suit various loads. 

Another form of sliding brake, also designed by the author, 
is shown in Fig. 942. In this design the strap b, is given such 





Fig. 942. 

tension t, by means 'of the screw e 7, and lever c, as to hold the 
load from descending ; a rubber spring being introduced at 7. 
If the load is to be lowered, the clamp e, is loosened, but is 
again tightened on ceasing. When hoisting, the tension t at 2" 
is readily overcome. This is in reality a form of running ratchet 
gear, and as shown it is made with a strap of wedge section, the 
angle 6 being 45°. The wedge portion is made of wood on iron 

at least 0.20I increased by -^ which when used to multi- 

sin — 
2 

ply the value of _/ 0( , requires a very small force to overcome 
the tension t. 



?3o6. 

Chain Brakes. 

Chain may be 
used as the ten- 
sion organ in 
brakp construc- 
tion, in which case 
it is generallF 
lined with blocki 
of wood, as in Fig. 
-— 943. The tensions 
T and t, to be 
given to the two 
parts of the cha:::. 
are readily oV*- 
tained from for- 
mula (312). The 
ratio of chain 
pitch length I, to 
the pulley radius 
r, is increased be- 
cause of the use of the wooden block. When / = J.^ rand the 
arc of contact is less than iSo° we have : 




Fig. 943- 



•=-r=(-i)' 



(316) 



For wood on iron we may take y ^ 0.3 (see section 193). 
This gives : 

T 

p = -7- = I . 1' = 2.35 ; also 



T_ 
~P'' 



"p — ' 1-35 



= 1.74 



and -g- =: T — I = 0.74, or / ^ 0.74 P. 

These proportions should not be strictly followed for heavy 
brakes such as in Fig. 939, as such should be determined for 
each case. 

?3o7. 
iNTERNAi, Strap Brakes. 

Strap brakes maj' be used in internal pulleys, in a manner 
similar to the internal ratchet gear of Fig. 711, for example. 
The outside of the strap then acts upon the inner surface of the 
pulley, the strap being subjected to compression instead of ten- 
sion,* thus becoming a pressure organ, a subject treated more 
fully in the following chapter. 




The pressure of the internal strap brake is of the same mag- 
nitude as with the external brake, but in the opposite direction, 
so that the previouslj' determined value of p from the forces T 
and /, may be used. Fig. 944 shows three forms of such brakes, 
these being used for friction couplings, and not in hoisting 
machinery (see Fig. 449). Fig. 944 a, is Schurman's friction 
coupling, t The brake lever c, acts by means of a wedge 4, 
upon one end of the strap. The other end of the strap is held 
by a pin 3, to the member d, whic"h is to be coupled to a by 
means of the strap b. The lever c, is also pivoted to the mem- 
ber d. For the forces T and /, we may use formula (^39), and 
since o< is nearly = 2 -, or say = 6, we have ior f = 0.1 the 
valuey^ ex = 0.6, which from the table of ? 264 gives p ■^. 1.S2, 
and T =1 2.22, whence t := 1.22 P. The strap must be released 
by the action of a spring. 

Fig. 944 b, is Adyman's coupling, j which is made with a 
heavy cast iron ring. The ring b, is made in halves, b' and i", 
fitted with projections 4' and 4'^ which engage with an interme- 
diate sheave keyed on the shaft. 



♦See Theoretical Kinematics, p. 167 ; p. 54S. 

t Zeitschrift des Vereins Dentscher Ingcnuiere, Vol. V. p. 301, 

X Made by Bagshaw & Sons, Batley, Yorkshire. 



2l6 



THE CONSTRUCTOR. 



The levers c' and c" have a common axis at 5, and when 
separated by a wedge at 6, they press upon the ends of the ring 
at 3' and 3". A pin at 7, keeps the levers from sliding in the 
direction 7 . I, as well as the ring b' b" . 

The coupling shown in Fig. 944 f , acts both ways, as an inter- 
nal and external strap brake, and is used on a shaping machine 
by Prentiss. * The steel strap b, is covered with leather. When 
the arms c' c" are drawn together it acts as an external strap 
on the pulley a" , and when they are forced apart it becomes an 
internal strap iu the pulley a' . The arms c' c" are carried on 
sleeves and are rotated to or from each other by a screw action. 



force" of the pressure organ serving to retain it within the 
desired limits. Canals are merely conduits of larger dimen- 
a 1) c 



CHAPTER XXIII. 

pressure organs considered as machine elements. 

Various Kinds of Pressure Org.^ns. 

In distinction from the various kinds of tension organs which 
have been considered in the four preceding chapters, there 
exists another group of machine elements of which the sole or 
principal characteristic is that they are capable onlv of resist- 
ing forces acting in compression. This group includes fluids, 
both liquid and gaseous, whether limpid or viscous, such as : 
Water, oil, air, steam, all pasty substances, clay, molten metals ; 
also granular materials, all kinds ^i grain, meal, gravel, etc. 
In all these materials the principal feature lies in the fact that 
the particles are subdivided to such an extent that they can be 
separated from each other by a very small force, while on the 
other hand they are capable of opposing more or less resistance 
to compression, this resistance in many instances, as, for exam- 
ple, in the case of water, almost equalling that of metals. These 
materials may be used as machine elements in a great variety of 
ways, and in the following discussion they will be included 
under the general title of Pressure Organs. Like their coun- 
terparts the tension organs already discussed, they are used 
largely for the transmission of motion in various manners, but 
are of still greater importance on account of the great variety 
of physical conditions in which they appear. 

\ 329. 

Methods of Using Pressure Org.-vns. 

The distinction which has been made between tension .and 
pressure organs enables various points of contrast and compari- 
son to be made as regards the methods of ulilizing them, and 
pressure organs may be divided in the same manner as tension 
organs (see \ 262) into standing and running organs. These 
divisions have but little practical application in this instance, 
but the three following subdivisions in 'i 262, viz. : Guiding, 
Supporting (/. c , raising or lowering), and Driving are here 
applicable also. We may therefore distinguish pressure organs, 
when considered as machine elements, into the following 
classes : 

1. For Guiding. 

2. For Supporting (including raising and lowering). 

3. For Driving. 

These various methods of action may he used either separately 
or in combination, and are found in most varied forms in many 
machine constructions. The great variety of possible combina- 
tions makes it desirable for a general view of the entire subject 
to be taken before discussing details. 

I 310. 

Guiding by Pressure Organs. 

In order to use a pressure organ for guiding, /. e., to compel a 
more or less determinate succession of motions, it is necessary 
to use also two other machine elements formed of rigid mate- 
rials. These latter are for the purpose : 

(?, Of resisting the internal forces of the pressure organ 
and keeping it within the desired limits. 

b. Of connecting the pressure organ with the external forces 
to be received and opposed. 

Tubes, Conduits, Canals. — The tube a. Fig. 945, limits the 
toundary of the particles of the pressure organ, and retains it 
in the desired form and controls its direction. A bend in a 
tube corresponds to a pulley around which the pressure organ 
is bent, and thus has its direction changed. Even when no 
change of direction is made, the tube is necessary to oppose re- 
sistance to the particle of the pressure organ, and hence at 
every section it must offer resistance to tension as well as com- 
pression. Conduits, or channels, as at b, are tubes with one 
side left open, the force of gravity or the so-called "living 




Fig. 945- 

sions, as at e, and natural streams of water often serve the pur- 
pose. 

Driving Organs, Pistons and Cylinders. — The bodies by 
means of which the pressure organ is connected with the exter- 
nal forces and resistances with which it is intended to act 
mechanically may be called generically. Driving Organs, and 
are very varied in character. Among these are movable recep- 
tacles, also moving surfaces or moving conduits (as in turbines), 
and also moving pistons in tubes or cylinders. A piston serves 
to oppose the stress in the pressure organ in the direction of its 
motion, while the walls of the tube oppose their resistance at 
right angles to the direction of motion. The inclosure in which 
a piston acts is called, in general terms, the cjdiuder, and details 
of construction will be given hereafter. The principal types 
will here be considered briefly. 

A complete working contact between piston and cylinder can 
only be obtained when both surfaces are alike, and this is only 
geometrically possible with three forms of bodies ; /. e., prisma- 
tic bodies, bodies of rotation, and spirally formed bodies. Of 
these the prismatics are most useful, and among the prismatic 
bodies the form most extensively used is the cylinder. 

The fit of a piston in its cylinder, entirel}' free from leakage, 
is very difficult of attainment, and is rarely attempted in practice. 

In steam indicators the piston is very accurately fitted directly 
into the cylinder, but in most cases a practically satisfactory 
result is obtained by the use of some intermediate packing 
device. 







In many cases a soft packing of hemp or leather is used, Fig, 
946. At a is shown a piston with external packing, at b an 
internal packing. In these cases one entire end of the cylinder 
is open, the piston filling the entire cylinder and acting upon 
the inclosed pressure organ on one side, this constituting a 
single-acting position. At c and d are similar double acting 
pistons. Pistons of the forms shown in a and b are sometimes 
called plungers, and the shorter inclosed pistons, as c or d, are 
also called piston-heads. At f is a double-acting piston used in 
connection with a rod and stuffing box, the rod being connected 
with external mechanism, and the stuffing bo.x made either with 
external or internal packing, as indicated at i and \' . In many 
instances pistons are made with openings which are fitted with 
valves, and hence may be called "valved" pistons, while those 
here shown are termed closed or solid pistons. 

The tightness of the packing is usually produced by the appli- 
cation of some external force, but in the so-called forms of self- 
acting packing the necessary pressure is supplied by the con- 
fined fluid. This is shown in the following illustrations. 

a b 





LI 



*See Mechanics Feb., 18 



Fig. 497- 
Fig- 947 <^ and b, Cup packing for piston or stuffing box ; metal 



THE CONSTRUCTOR. 



217 



packing, usually for pistons, but also used in stuffing boxes. 
The fluid in all three cases enters behind the packing rings and 
tightens the joint in proportion to the increased pressure. 

In the class of self-acting packing may also be included the 
various forms of liquid packing, some of which are given in 
Fig. 94S. The forms at a and b are practically plungers, while 

a b 




X 




Fig. 948 

in many cases an ordinary packing lias its tightness increased 
by a layer of water or oil upon the piston, as shown at c. 

Another variety occurs when the connection between cylinder 
and piston is made by means of a membrane or diaphragm, as 
in Fig. 949. 





Fig. 949. 

These are among the oldest forms of transmission organs, 
but are practically true pistons in principle and action. At a is 
a single diaphragm, known as the monk's pump ; b is the so- 
called 'bellows" form ; r is a series of flexible metal diaphragms, 
usually of steel, brass or copper, used for pressure gauges or 
other similar purposes involving but little movement. At d is 
the so-called " bag" pump, in which the liquid does not come 
in contact with either cylinder or piston, but is confined within 
a flexible bag. 





Fig. 950. 

Another class of pistons is that in which a tight packing is 
not attempted, these usually being used only for air. Fig. 950 a 
shows a deep piston with grooves formed in it, the fluid endea- 
voring to pass the piston in the opposite direction to the motion 
of the latter, becomes entrapped in the grooves, and before it 
can pass, the direction of motion is changed and this actiou 
reversed.* ht b\& a piston with a brush packing, used for a 
blowing cylinder at Sydenham. In this class of pistons we may 
also include floats which rise and fall with the motion of the 
liquid. vSuch floats are shown at c and d, the former being open 
and the latter closed. A solid block may also be used for this 
purpose, if its weight is nearly counterbalanced b\' another 
weight. 

Details of piston and cj'linder construction will be given in 
Chapter XXVI. The corresponding machine elements to pis- 
tons in tension organs will be found for ropes in Figs. S25-S26, 
and for chains in Figs. S31 to 834. The change of direction from 
compression to tension dispenses with the necessity for a 
cylinder. 



Guide Mech.\nism for Pressure Organs. 

The combination of a pressure organ and its accompanying 
guide mechanism forms a pressure transmission system. Fx- 




a b c d e 

Fig. 931. 

amples of such systems are given in outline in Fig. 951. At a 
is an arrangement for raising the load Q vertically. The 
plungers b and d are of the same diameter ; the pressure on b 
must be the same as Q, neglecting friction. The column of 
water is the same diameter as the plungers, and the direction is 
changed an angle of 120°. It is desirable that distinguishing 
names should be given to the various arrangements. If we 
compare these with the corresponding parts in tension organs, 
Fig. 7S4 and Fig. 7S5 a, we may properly call such an angle 
transmission a hydraulic pulley, or water pulley, but a still bet- 
ter name is the "hydraulic-lever" or "water- lever," which will 
be hereafter adopted. 

At b is shown a free water-lever. The plungers b and d are 
equal in diameter, the load Q is supported on two columns of 
water, hence, if friction is neglected, the force on each plunger 
will be I2 Q, the angle of change of direction is 180°. 

At c is a combination of case a with case b. The plungers 
^1, b,, b^, are of the same diameter, and the load O is supported 
on these columns. These three cases correspond in principle 
with the similar cases a b c of Fig. 7S4. Since the three plungers 
ij, b.,, A), of case c all exert the same force, they may also be 
made to give the same result when made as shown at d, or if 
the three plungers are combined in one, form e is obtaiued. 
The latter form is well known in practice as the hydraulic 
press. The principle iuvolved in all these devices is the same 
as appears in the various pulley systems of tension organs. 

A comparison of Fig. 95 1 a with e shows that the same prin- 
ciple exists in both, and case a may be considered as a water- 
lever of equal arms, and case if as a lever of unequal arms. 




Fig. 952. 

The water-lever has been used in more or less complete de- 
vices for balancing the weight of pump rods in deep mine 
shafts. Fig. 952 shows Oeking's water couuterbalance.f The 



* See Weisbach, Vol. III., Part 2, ? 410. 



t Zeitschrift Deutscher Ingenieure, 18S5, p. 545. Oekin§ 
the device « ^ au accumulator. 



incorrectly call 



2l8 



THE CONSTRUCTOR. 



pump rod is carried on the two plungers d^ d.,. and its weight 
counterbalanced by the weighted plunger and cylinder a~b. 

In the Emery scales and testing machines water-levers of 
unequal arms are used in connection with metallic diaphragms. 



■^'■^''^^-^^''^■'^■ ^-'^■'■^^yy'^yCi^^i^^ 





d 

Fig. 953- 

Fig- 953 shows a combination of two hydraulic levers, each of 
the form of Fig. 931 a. The weight Q travels in a straight line, 
being kept parallel by the four equal plungers bib.,bo,b^, and 
crossed pipe connections. This construction is similar to the 
cord parallel motion of Fig. 7S4 d. 

In all of the devices described the rigid body is guided by the 
motion of the pressure-organ. It must be remembered that 
motion is merely a relative term, and the rigid body may move 
through the fluid. An example of the latter is the rudder of a 
vessel, which acts in one plane ; or in the case of the Whitehead 
torpedo .several rudders are used, guiding the torpedo in any 
direction. 

I 312. 

Reservoirs for Pressure Organs. 

Reservoirs are used in connection with pressure organs in 
order to enable a number of applications to be operated collec- 
tively, and also to enable the pressure to be stored for subse- 
quent service, and in this respect they correspond to the variovis 
forms of winding drums used with tension organs, and shown 
in Fig. 787. The following illustrations will show the use of 
such reservoirs. 

Fig. 954 shows a tank for use with petroleum distribution, as 




Fig. 954. 



used in the American oil fields, and more recently in the oil 
district of Baku. The oil wells are at «,, a.^, a^, and the oil is 
forced to the elevated reservoir at c by pumps. From the reser- 
voir the oil flows to the point of shipment d, and the supply is 
gauged by the fltictuations of level in the tank.* 

The reservoirs used in connection with the water supply of 
cities are similar in principle. Where the configuration of the 
land demands it, the pipes are run in inverted siphons connect- 
ing intermediate reservoirs. An illustration of this arrange- 
ment is given in Fig. 955, which shows the waterworks system 
of Frank^urtam-Main designed by Schmick. 




Fig. 955. 

The highest spring is at «,, Vogelsberg, and the next at a.,, 
Spessart. These both deliver into the reservoir r,, c,, at Aspeu- 
hainerkopf. The next reservoir is at <:.„ Abtshecke, from which 
the water flows through b^ to the reservoir t:^ and c\, from which 
the city is supplied. The elevations above sea level are given 



in metres. The flow between the various reservoirs is controlled 
by suitable valves. f 

Small tanks are iu very general use at railway stations ; and 
the various ponds and mill dams used in connection with water- 
wheels are other examples. In many cases the water ways are 
large enough to serve as reservoirs also, as in the case of canals. 

Natural reservoirs are found iu the case of many mountain 
lakes, the Swiss lakes affording many numerous instances.f 
Such basins are also formed artificially by constructing dams- 
across narrow outlets, and so storing the water for use. Note- 
worth)' examples found in France, the basin at St. Etienne, 
formed by damming the river Furens, beuig over 164 feet {5'> 
metres) deep. J 

Water may also be stored in accumulators at high pressures 
from 20, 50, as high as 200 atmospheres, and can then be used 
for operating hydraulic cranes, sluice gates, drawbridges, etc. 
These accumulators may be considered as a form of releasing 
ratchet mechanism (see I 260). To this class of mechanical 
action also belongs the system, used in the Black Forest, by 
which the streams are temporarily dammed and then suddenly 
released in order to float the logs down with the sudden rush of 
the current. 

Iu using high pressure water transmission it is sometimes 
desirable to transform a portion to a lower pressure in order to 
operate a lower pressure mechanism, or by a reversal of the 
same principle, to convert a lower to a higher pres.'iure. This 
can be done by uieans of the apparatus devised by the author,, 
and shown iu Fig. 956. \\ 







Q 


f 




a 




rL 




M 





Fig. 956. 

This is a form of hydraulic lever of unequal leverage, but 
is diff'erent from those shown iu Fig. 95'- Referring to Fig. 
956(7, the high pressure water is delivered at a, aud connected 
with the lower pressure water a^ by means of the plungers b, b„ 
the latter being iu one piece of two different diameters. The 
difference in pressure, neglecting friction, will be inversely as 
the areas of the two plungers, or if they are of circular section, 
inversely as the squares of their diameters. In this case the 
lower pressure then acts in the cylinder c upon the plunger d. 
The action of this arrangement may be considered as if the 
plungers b and />, were upon the same axis and rigidly con- 
nected, and the leverage compounded in a manner similar to 
that of the rope crane of Fig. 792(7; this comparison being 
more clearly shown by referring to Fig. 956 b, This device may 
also be used as a supporting hydraulic lever, similar to Fig. 951 e. 
If a communication is made between the two different water 
columns, as shown in Fig. 956 r, the pressure wdll be equalized. 
This gives a differential hydraulic lever similar in principle to 
the Chinese windlass of Fig. 790 a, or the Weston Differential 
Block of Fig. 796 e. , 



* A system of this .sort was built in 1SS7 from Balcii to Batoum on llie 
Black Sea. The length of line is 1005 kilometres (603 miles), 6 in. diameter, 
and the reservoirs 3000 feet above sea level. 



t A large inverted siphon is formed by the new Croton Aqueduct, which 
passes under the Harlem River at a depth of 150 feet below the surface of ther 



river, and a tunnel of loV^ feet iu diameter driven through the solid rock. 
See Mechanics. Nov , 1SS6, p. 24Z. 

I This is examined in detail in a memorial on the better utilization of 
water, published at Munich in 1883 by the German Society of Engineers and 
Architects. 

g For further discussion of this subject the following reft-rences may be 
consulted ; Jaubert de Passa, Recherches sur les arrosages chez les peuples 
anciens, Paris, 1S46; Ditto, Memoire sur les cours d'eau et les cananx 
d'arrosages des Pyrenees orientates; Nadault de Buffon, Cours d'agriculture 
et d'hydraulique agricole, Paris, 1853-1S5S; Ditto, Hydraulique agricole, ap 
plication des canaux (I'irrigation de I'ltalie septentrionale. Paris. 1S61-1S63 ;. 
Baird-ftniyth, Irrigation in Southern India, London, 1856 ; Dupuit, Traite de 
la conduite et de la distr. des eaux, Paris, 1865 ; Scott-Moncrieff, Irrigation in 
Southern Europe, London, 1868; Linant de Bellefonds Bey, Memoire sur les 
principaux travaux d'utilite publique en Egypte etc., Paris, 1873 ; Krantz, 
Etude sur les murs de reservoirs. Paris, 1870 ; F. Kahn, TJeber die Thalsperre 
der Gileppe bei Verviers, Civil ingenieur, 1S79, p. i; also an article by 
Charles Grad in " la Nature," 1S76, p. 55 ; also a t)rief article by the author 
'■ LTeber das Wasser," Berlin, 1S76. 

II See Glaser's Annalen, 18S5, Vol. XVIL, p. 234. 



THE CONSTRUCTOR. 



219 



The opposite extreme to a high pressure accumulator is found 
iu those pools or receptacles of water far below the natural sea 
level, such as are found iu mines, and iu the polders or drainage 
pools of Holland, Lombard^', and parts of Northern Germany. 

Reservoirs are not confined to use with liquids. Examples of 
Other fluids are found iu the gasometers of gas works, in the 
receivers for compressed air, so e.xtensively used in mining and 
tunneling, and iu making the so-called pneumatic foundations. 
Smaller reservoirs are found iu the air-chambers on pumping 
machinery, and the like. 

The sewage system of Berlin, designed by von Hobrect, con- 
sists often drainage pits, with the water level below the natural 
level, an-auged on the so called radial system. The sewage is 
pumped from these pits and delivered by means of pipes to 
sewage farms at a distance from the city. 

Negative receivers, so-called, may be used for air, as iu the 
case of the coining presses of the English mint, where a vacuum 
chamber is used to receive the air already used for driving the 
machines, and kept pumped out by steam power. The venti- 
lating apparatus for mines also often contains such negative 
reservoirs for air. 

Reservoirs are also used for granular materials, such being 
extensively used in connection with grain handling machinery. 

A steam boiler may be considered as a physically supplied 
reservoir, as well as a physical ratchet system (see \ 260). A 
combined physical and chemical reservoir is found in the elec- 
trical accumulator, which ma^' properly be called a current- 
reservoir. A combined physically and mechanically operated 
negative reservoir is found in the various forms of refrigerating 
machines. 

A modern application of pressure organs, and one which is 
rapidly extending in use, is that of the distribution of power in 
cities. Followiug the impulse given by the introduction of the 
high pressure water S3'stem of Armstrong, the use of gas in 
motive power eugines by Otto followed, and many other methods 
of meeting the problem have been applied. 

In long distance transmissions of this sort, special reservoirs 
are often used, in which force may be stored, so to speak, and 
from thence distributed in a manner similar to the ring trans- 
mission system for rope (see § 301). In this method the pres- 
sure organ after use is returned to the reservoir to be compressed 
and used again, or it may be used as in the line transmission 
and allowed to escape at the end of the line.''' 

The following cases are given as applications of pressure 
organs in long distance transmission : 

1. The London Hydraulic Power Company distributes 300 
H. P. by means of water at a pressure of 46 atmospheres (675 
pounds). A similar and earlier installatiou is iu use at Hull. 

2. The General Compressed Air Company distributes power 
by means of air at a pressure of 3 atmospheres {45 pounds) in 
Leeds and Birmingham. The system is an open liue, and 1000 
H. P. are used in Leeds, and 6000 H. P. in Birmingham. f In 
Paris the Compagnie Parisienne de I'air comprim^, procedes 
Victor Popp, distributes power from three stations iu quantities 
varying from a few foot pounds up to 70 or So H. P., a total of 
some 3000 H. P. The use of compressed air appears to be 
destined to a widely extended use for this purpose. 

3. The distribution of power in New York by means of steam 
mains is extensive and well known. 

4. The vacuum system is used also iu Paris by the Societe 
auonyme de distribution de force a domicile. This is au open 
line transmission, operating in 1SS5, about 200 H. P. 

5. Transmission by highly superheated water has been used 
in Washington, by the National Superheated Water Co., dis- 
tributing heated water at pressures from 26 to 33 atmospheres 
(400 to 5oo pounds), the water being converted into steam at the 
point of utilization. 

6. The distribution of power by means of gas holders has 
already been referred to, and the distribution by electric cur- 
rents is rapidly being developed. 

I 313- 
Motors for Pressure Organs. 

The methods of applying pressure organs to the development 
of motive power are even more varied as in the case of tension 
organs. For this reason a general view of the subject will be 
taken in order to obtain a classification which will simplify the 
discussion. The main distinctions are those of tlie character of 
the motion of the mechanism, and of the method of applying 
the pressure organ to the motor. 

The great difference in the character of the motion of the 



mechanism lies in the fact that it may be either continuous or 
intermittent, so that the motor may be either : 

A running mechanism, or 

A ratchet mechanism (compare \ 260). The ratchet pawls for 
pressure organs are the various forms of valves (see Chapter 
XXVI). 

The various forms may also be classified according to the fol- 
lowing important distinctions based ou the method of driving. 

The pressure organ may drive, or 

It may be driven, or 

The impelling mechanism may itself be propelled. 

There is also a third distinction to be made, uamely, whether 
the pressure organ acts merely by its weight, or whether it acts 
by its living force of impact. This last distinction cannot be 
sharply observed in practice, but is especially to be considered 
in discussing the theory of action of the various machines. 

In the following pages the various applications will be shown 
in a manner similar to that employed in '/. 262 for tension organs, 
following the system of classification outlined above, aud be- 
ginning with running mechanism as the simpler of the two 
great divisions. 

A. RUNNING MECHANISM FOR PRESSURE ORGANS. 

I 3M- 

Running Mechanism in which the Pressure Organ 
Drives by its Weight. 

With a few unimportant exceptions the motors of this class 
are operated by liquids, which at moderate velocities practically 
follow the laws of gravity. 

Iu Fig. 957, a is an undershot water-wheel, and 5 is a half- 




Fig. 957. 

breast water. The water is guided in a curved channel and the 
buckets are radial, or nearly so. The wheel is so placed that 
the buckets pass with the least practicable amount of clearance 
over the curved channel. At c is shown a high-breast wheel, 
and at d an overshot wheel (compare §47). In these latter 
wheels the buckets are so shaped that they retain the water in 
the circular path, being closed at the sides also, while on 
account of the moderate pressure they are left open above. At 
e is shown the side-fed wheel of Zuppinger. 

Fig- 958, a is an endless 
chain of buckets, and b a 
similar arrangement, using 
disks running with slight 
clearance in a vertical tube. 

In the wheels shown in 
Fig. 957 the water acts on 
the wheel much in the same 
manner as a rack acts when 
driving a pinion, and iu this 
sense a water wheel may be 
considered as a gear wheel. 

When the water acts only 
by gravity these construc- 
tions are only practical when 
the wheel can be made 

larger in diameter than the p „ ._g 

fall of water, and where 

small diameters must be used the arrangements of Fig. 958 are 
available. Very small wheels acting under high pressures may 
be employed by making use of the so-called " chamber wheel 
work," X of which some examples are here giveu. 





* See a paper by the author in Glaser's Annalen, 1885, Vol. XVII., p. 226. 

t See Lupton and Sturgeon, Compressed .-Vir vs. Hydraulic Pressure, Leeds, 
1886; Sturgeon, Compressed Air Power Schemes,' London, 1S86 ; also The 
Birmingham Compressed Air Company, Birmiugham, 1S86. 



Fig. 959(1 is the Pappenheim chamber wheel train. In this 
the tooth contact is continuous, the teeth being so formed that 
the continuous contact of the teeth at the pitch circle prevents 



\ See Berliner Yerhandlungen, 1S6S, p. 42. 



220 



THE CONSTRUCTOR. 



the water from passing, while the points and sides of the teeth 
make a close contact with the walls of the chamber. The 
downward pressure of the water enters into the spaces between 
the teeth and drives both wheels. The axes of the wheels are 
also coupled by a pair of spur gear wheels outside the case, 
thus insuring the smooth running of the inner wheels. This is 
the oldest form of chamber train mechanism known, and can 
also be used as a pump, operating equally well in either direc- 
tion. Fig. 959 b is Payton's Water Meter, with evolute teeth. 
The flow is intermittent, but one contact begins before the 
action of the previous one ceases. 

Fig. 959 c is Eve's chamber gear train. The ratio of teeth is 
I to 3, and the flow is also intermitteut. The theoretical volume 
of delivery for all forms of chamber gear trains, whether con- 
tinuous or intermittent in deliver3-. is practically equal to the 
volume described by the cross section of a tooth of one of the 
two wheels for each revolution. 

Fig. 959 rf is Behren's chamber train. In this case each wheel 
has but one tooth, as is also the case with Repsold's train (de- 
scribed hereafter), and the gears belong to the class of disc 
wheels or so-called "shield gears " (see g 2ii). This arrange- 
ment possesses the great advantage of offering an extended sur- 
face of contact at the place between the two wheels where, in 
the previous forms, there is but a line contact. This permits a 
sutficietit degree of tightness to be obtained without requiring 
the parts to press against each other. Behren's chamber gear 
makes an excellent water motor if the impurities of the water 
are not sufficient to injure the working parts. 

The flow of water through chamber gear trains is not uni- 
form, and the inequality of delivery increases as the number of 
teeth in the wheels is diminished, hence they should be driven 
only at moderate velocities when used as motors, in order to 
avoid the shocks due to the impact of the water. 

§315- 

Rdnning Mechanism in which ths Prbssure Organ 
Drives by Impact. 

In driving running mechanism by impact, fluid pressure 
organs, both liquid and gaseous, may be used, as will be seen 
from the following examples. 





Fig. 960. 

Fig. 960 a; is a current wheel, or common paddle wheel. The 
paddles are straight, and either radial, or slightly inclined 
toward the current, as in the illustration. The working contact 
in this case is of a very low order. 

Fig. 960 b is Poncelet's wheel. The buckets run in a grooved 
channel, and are so curved that the water drives upwards and 
then falls downwards, thus giving a much higher order of con- 
tact. At c is shown an externally driven tangent wheel. The 
buckets are similar to the Poncelet wheel, but with a sharper 
curve inward. The discharge of the water is inwards, its living 
force being expended. At d is an internally driven tangent 
wheel, similar to the preceding, but with outward discharge. 
The form shown at e is the so-called Hurdy-Gurdy wheel. The 
water is delivered through curved spouts, and this form is prac- 
tically an externally driven tangent wheel of larger diameter 
and smaller number of buckets. This wheel, from a crude 
makeshift, has become one of the most efficient of motors.* 

Wheels with inclined delivery as made in the forms shown in 

a 1) c 




Fig. 961. 
Fig. 961. At a is shown a crude form, used on rapid mountain 



* This is the Pelton Water Wheel, built in sizes as great as 300 H. P. See 
Mining- and Scientific Press, 18S4, p. 246. and 1S85, p. 21. This wheel is built 
in Zurich, by Escher, Wyss & Co., with a casing, and used for driving 
dynamos. 



streams as a simple ex-pedient, but of low efficiency ; b is the 
Borda turbine, consisting of a series of spiral buckets in a bar- 
rel shaped vessel ; c is tne so-called Danaide, the spiral buckets 
being in a conical vessel, this form being mostly used in France.! 

In the wheels which have been shown in the preceding illus- 
trations from Fig. 95S, the living force of the water acts by 
direct impact through a single delivery pipe. 

The following forms difi^er from the preceding, in that the 
water acts simultaneously through a number of passages around 
the entire circumference of the wheel. This form gives the so- 
called hj'draulic reaction in each of the inclosed channels, and 
hence wheels of this class are commonly called reaction wheels, 
or reaction turbines.J 




Fig. 962. 

Fig. 962 a is Segner's wheel, the water entering the vertical 
axis and discharging through the curved arms ; b is the screw- 
turjpine, entirely filled with water ; c is Girard's current turbine, 
with horizontal axis, and only partially submerged ; of is Cadiat's 
turbine, with central delivery, and e is Thomson's turbine with 
circumferential delivery and horizontal axis, the discharge being 
about axis at both sides. In all five of these examples the 
column of water is received as a whole, and enters the wheel 
undivided until it enters the wheel ; in the following forms the 
flow is divided into a number of separate streams. 




Fig. 963. 

Fig- 963 ci is the Fourueyron turbine, acting with central 
delivery ; the guide vanes are fixed and the discharge of the 
water is at the circumference of the wheel ; (5 is a modification 
of the Fourueyron turbine, the water being delivered upwards 
from below, and sometimes called Nagel's turbine; c is the 
Jouval or Henschel turbine, the guide vanes c being above the 
wheel, which is entirely filled by the water column ; d\s Fran- 
cis' turbine, .with circumferential delivery through the guide 
vanes c* ; e is the Schiele turbine, a double wheel with circum- 
ferential delivery and axially directed discharge. In the latter 
three forms a draft tube may be used below the wheel, to utilize 
that portion of the fall, as indicated in forms c and d. 




Fig. 964. 

For gaseous pressure organs, of which wind is the principal 
example, some forms are here given. Fig. 964 a is the German 
windmill, with screw-shaped vanes ; b is the Greek and Anato- 
lian windmill, with cup-shaped vanes. Both forms are similar 
in action to the above described pressure %vheels. At c is shown 
the so-called Polish windmill, with stationary guide vanes ; || 
(/is Halladay's windmill, made with many small vanes, which 
place themselves more and more nearly parallel with the axis 
as the force of the wind increases, the rudder r, keeping the 
wheel to the direction of the wind. The extreme position of the 
vanes is shown at e- Anemometers and steam turbines are 
examples of wheels in which other pressure organs than wind 
are used. 



, Sccti 



on 4, 



t See Wei.sbach-Hernnian, Mechanics of Engineering, Part II., 
p. 55S., 

X This use of the term reaction is hardlv desirable for this construction, 
nor is the proposed name of " action turbine," and the name "pressure 
turbines " is to be preferred. 

§ This form is well made by J. M. Voith.of Heidenheim, Wtirtemberg. 

II Recueil des Machines avantageuses. Vol. I.. No. 31, 1699, also from thence 
shown in Henning's Sammlun<jvon Machinen und lustrumenten, Niirn- 
berg, 1740. 



THE CONSTRUCTOR. 



221 



I 316. 

Running Mechanism in which the Pressure Organ is 
Driven against the Action of Gravity. 

Running mechanism for the purpose of elevating liquids, and 
especially for lifting water, are of very early origin, and the 
various machines for this purpose form the very oldest of 
machine inventions. 



there is no necessity for distinguishing in classification between 
them as pumps for liquids or for gaseous fluids. Fig. 967 c is 




Fig. 965. 

Fig. 965 a is a bucket wheel, the vessels on the circumference 
lifting the water ; this is driven by the power of men or animals, 
or in many instances by a current wheel (as in Fig. g6o«).* At 
b is the Tympanon of Archimedes, used down to modern times, 
the sections deliver the water through openings into the axis; 
c is a paddle wheel, only adapted to raise the water a small 
height, much used in the polders of Germany, Holland and 
Italy. The paddles are made either straight, or curved, or 
sometimes slightly crooked at the end.f At d is the Archime- 
dian screw, which, wheu placed at an angle as shown, is well 
adapted to elevate water. The Archimedian screw is exten- 
sively used in all positions for the granular and pulverized 
materials, in which cases the outer cylinder is omitted and a 
stationary channel substituted, as shown at c, in Fig. 965 e, and 
if the transportation of material is in a vertical direction the 
screw is entirely surrounded by a stationary tube. A still later 
form is made with a wire spiral, by Kreiss of Hamburg. 




Fig. 966. 

Fig. 966 a is the spiral pump, in which the screw of Archi- 
medes is replaced by a channel formed in a plane spiral. In 
this form the inclosed air becomes compressed by the speed of 
revolution of the mass, and the water can be forced quite a con- 
sijierable height, j Fig. 966 ^ is a conical spiral pump called 
after its inventor, Cagniard Latour, a Cagniardelle. The Cag- 
uiardelle is usually placed entirely in a trough, but the illustra- 
tion shows how the end of the spiral may be modified so as to 
require no enlargement of the delivery channel. The diameter 
of the cone is adapted to the height to which the water is to be 
lifted. The Cagniardelle may also be used as a blower, the in- 
closed water driving the entrapped air before it. 

The chain and bucket devices already shown in Fig. 958 as 
motors are also well adapted to drive the pressure organ, and 
are in practical use in numerous modifications. Fig. 958 a is 
extensively used in dredging machinery, grain elevators and 
the like, and Fig. 958 b is nmch used for lifting water. 

The various forms of chamber gear trains already described, 
give by inversion corresponding forms of driving mechanism, 
some examples of which are here given. 

Fig. 967 a is Repsold's pump ; each wheel has one tooth, the 
profiles Ijeing formed as described in 5 207 ; b is Root's blower, 
the wheels having two teeth each, and the action being the 
same as the Pappenheim machine. Fig. 959;?. This device has 
been very extensively used as a blowing machine. Since the 
action of these machines in drawing air against pressure is simi- 
lar to that of lifting water against the resistance of gravitj-, 



* Large wheels of this sort have been in use in Syria for many centuries, 
as at Orontes, north of Damascus, l^he town of Haniatli, of 40,000 inhabi- 
tants, receives its water supply from twelve such wheels. 

1" A recent installation of such paddle wheels has been made at Atfeh. on 
the Mahmudieh Canal, in Egypt. Eight wheels 32.8 feet diameter, each 
driven by a separate steam engine lifting water from the Nile 8^ feet to the 
canal. The eight wheels deliver 115,000,000 cubic feet in 24 hours. See 
Engineer, 1887, p. 57. 

X Such pumps, made by Klein. Schanzlin & Becker, at Frankenthal, deli- 
ver water from 2 to 30 feet, the revolutions being from 15 to 22 per minute, 
and diameters from ao to 70 inches. 




Fig. 967. 

Fabry's ventilating machine for mine ventilation, consisting of 
a double toothed combination chamber train, with unequal 
duration of contact. Root has also used the form shown at d, 
which has unequal contact duration, and which has since been 
made by Greindl as a pump.S 




Fig. 968. 

Greindl also makes the form shown iu Fig. 968 a, with gears 
of one and two teeth, and rightly claims it to possess the advan- 
tage of a greater freedom from leakage. The form shown at b 
has been used by Evrard as a blower, but it does not differ iu 
principle from a. Baker's blower, shown at c, is a triple cham- 
Ijer train, also used by Noel as a pump. 

It has already been stated that Behren's pump. Fig. 959 d, has 
also been used as a steam engine. As long ago as 1S67 a steam 
fire engine has been constructed by putting two of these 
machines on the same axis, one being driven by steam, the other 
forcing the water. 

Chamber gear trains may also be used to be worked in con- 
nection. Fig. 969 shows an arrangement iu which the chamber 




7m'/.^J>mii/:,:^//, 



Fig. 969. 



train A delivers water to a distant one B, driving the latter and 
receiving the discharge water from B through a return pipe to 
be used again. The combination forms a transmission system 
of the second order (see \ 26), and is similar to a belt or chain 
transmission. The loss in efBcienc}' in this device is not an un- 
important consideration. 

An important class of machines consists of those made with 
tension organs for transporting granular materials. For this 
purpose belts, chains, etc., are used, and when the transmission 
is horizontal, or nearly so, grain is successfully transported on 
wide belts. jl Another application is that of Marolles, using an 
iron belt, 40 in. wide, 0.06 in. thick, for transporting mud. 
Twelve such machines were used on the Panama Canal work, 
the distance being 200 feet, and the speed of the band 12 to 40 
feet, according to the nature of the material. Similar apparatus 
at the Suez Caual handled material at a cost of 7.6 cents per 
cubic yard. 

?3I7- 

Running Mechanism in which the Pressure Organ is 

Driven by Transfer of Living Force. 

The method of driving pressure organs by a transfer of living 
force is one which admits of numerous applications, as the fol- 
lowing examples show. 

Fig. 970 (7 is a centrifugal pump for moving liquids. The 
driving mechanism consists of the curved blades, which iu 



g The firm of Klein, Schanzlin & Becker, at Frankenthal, make a line of 
pumps similar to Fig 967 d, of a capacity of 1.77 to 177 cubic feet per minute, 
and discharge openings from i.iS to ii.8 ins. diameter. These are driven 
by belt and used beer-mash oil, acids, paper pulp, syrup, etc., as well as 
water. 

II An excellent transmission is in use at Cologne. See also Trans. Am. Soc. 
Mech. Engrs., Vol. VI., 1884-S5, p. 400. At the Duluth elevator a rubber belt 
50 inches wide, running 600 to 800 feet per minute, carries grain from 600 to 
goo feet horizontallv. A ^6" belt has carried 14,000 bushels per hour. 



222 



THE CONSTRUCTOR. 



many instances are made in one piece wilh the wheel itself, this 
adding to the efficiencs These pumps ha\e been most sue 
cessfuUy made by Gwjnne, Schiele Neut and Dumont among 




Fig. 970. 

others.* Centrifugal pumps have been successfully used as 
dredging machines for lifting wet sand, gravel and mud, in- 
stances among others being the North Sea Canal at Amsterdam, 
and the harbor at Oakland, California. 

Fig. 970 i is the well known fan blower u?ed everywhere for 
producing a blast of air, and acting by centrifugal force. When 
used <)s exhaust fan this is widely used in connection with 
suitable exhaust pipes for removing foul air, sawdust, and other 
impurities in workshops, as well as for the ventilation of mines. f 
At c is shown a form of spiral ventilator, known as Steib's ven- 
tilator; it is similar to some of the preceding forms, but is of 
limited application, and is better adapted for lifting water, a 
service to which it has been applied in the polders of Holland. 
At d is a centrifugal separator, a device of numerous applica- 
tions for separating materials of different specific gravity by 
centrifugal force. A notable example of this machine is the 
centrifugal separator for removing cream from milk. 

Another variety of machines for driving pressure organs by a 
transfer of living force, is that in which another pressure organ, 
either liquid or gaseous, is used instead of a wheel as the im- 
pelling mechanism. To this class belong the various jet devices, 
injectors, etc. 




Fig. 971. 

Fig. 971 ci is Giffard's injector in the improved and simplified 
form made by the Delaware Steam Appliance Co. In this case 
steam is used to drive a jet of water into a vessel already con- 
taining water under pressure. The jet of steam rushing through 
the nozzle b^ draws the water in by the suction tube b.^. and both 
pass through the mixing tube i,, and are discharged through 
the outlet tube b^ ; the outflow at b=^ provides for the relief of 
the discharge at starting, before the jet action is fully estab- 
lished. The regulation of the flow of steam is effected by a 
steam valve attached above i5,. At 4 is Gresham's automatic 
injector, which is so made that should any interruption occur 
in the supply of water at b.^, the suction action is automatically 
started, and the entering column of water is lifted again. This 
is done by the introduction of a movable nozzle b^ between b-^ 
and b^, wliich adjusts its position with regard to A3 according to 
the variations in pressure above and below. 

Fig. 972 is Friedmann's jet pump. The mixing tube i;, is 
divided into a number of sections, which permits a very free 
entrance to the water, and gives an excellent action ; b is 
Nagel's jet pump, used for lifting water from foundations by 
means of another jet of water. The entrance jet is at b-^, the 



* A recent installation of magnitude is that of five centrifugal pumps built 
by Farcot, of Paris, in 1887, for supplying the Katatbeh Canal in Egypt. 
The wheels are 12 ft. 6" dia , and each deliver 17,660,000 cubic feet in 23 hours, 
the lift varies from i to 12 feet. 

t Fans (or these purposes are made in great variety by J. B. Sturtevant. 
Boston, Mass. 





Fig. 972. 



the regulation is 



suction tube at b.^, and the mixing tube at b-^ 
effected by a valve at the end of b,,. 

Steam jets are also used to produce a blast of air, or com- 
pressed air may be used for the same purpose, as can also water 
under pressure. A reversal of the last mentioned arrangement 
occurs in Bunsen's air pump, in which a jet of water'is used to 
produce a vacuum. Recent devices for utilizing jet action are 
numerous. Among others, a jet of air has been used to feed 
petroleum into furnaces as fuel. Dr. W. ,Siemens proposed to 
carry the petroleum in the hold of a vessel in bulk, and substi- 
tute sea water, as it was consumed, in order to maintain the 
ballasting of the ship undisturbed. Granular materials have 
been handled by means of jet apparatus, usually impelled by 
compressed air, sometimes by water jets. 

An especial f^eature of jet pumps, and one which should not 
be overlooked, is that they act either by guiding the pressure 
organ stream, or that the driving action of the pressure organ 
stream itself produces a guiding action, and that the existence 
either of a reservoir or some external means 
<^ ^=(1 of driving must be presupposed. The use of a 
pressure organ in motion for driving mechan- 
ism, is in this respect similar to the so-called 
inductive action of an electric current. 

An example of pure guiding action is 
found in the "Geyser Pump" of Dr. W. 
Siemens, Fig. 973. The water is to be raised 
from a depth H , and the tube b is prolonged 
downward to a depth H^ below the sump 5'. 
The prolonged tube b-^ is open at the lower 
end, and in the bottom opening 7" an air tube 
c is introduced, and air is admitted at a pres- 
sure slightly under that of a column of water 
of height equal to H^. The air mingles with 
the water and forms a mixture in a^ which is 
lighter than water, and the air pressure is then 
capable of forcing the light mixture up to the 
surface. The lifting action is assisted by the 
expansion of the ascending air. Siemens 
found that it was possible to produce this 
action when //was equal to //,, that is, the 
specific gravity of the mixture of air and 
water =^ J4. 




-i-P' 



Fig. 973. 



? 31S. 



Running Mechanism 



IN which the 
Propelled. 



Motor itself is 



The third division, in which the motor itself is propelled in 
the liquid pressure organ, contains fewer varieties than the pre- 
ceding ones but is of the greatest importance since to it belongs 
the entire subject of marine propulsion. 





Fig. 974- 

Fig. 974 0' is the so-called " flying bridge," the current flow- 
ing in the direction of the arrow, causing the boats to swing 
across the stream, describing an arc about the anchor to wbicE 



THE CONSTRUCTOR. 



223 



they are held by a chain ; (!i, is a sail-boat, the sail beiug the 
driving organ transferring to the boat a portion of the living 
force of the current of wind. At c, is a steamboat with side pad- 
dle-wheels, and (/, a stern-wheel boat ; tf, is a screw propeller. 
A screw driven hy a steam engine pressing the water backward 
and the reaction of the water impelling the boat. Aty, is a 
so-called jet propeller, the reaction being produced by jets of 
water forced through tubes at the side of the boat, the water 
beiug driven b}' centrifugal pumps.* At^, is shown a current 
wheel motor. The side paddle wheels are caused to revolve by 
the action of the current, and by connection with a cable. or 
chain gearing (See Figs. 787 and 794) the boat is propelled up 
the stream. 

Direct acting reaction jets have been used for torpedo boats, 
using carbonic acid gas, but this method has been superseded 
by twin screw propellers driven by compressed air. Rockets 
and rocket shells are examples of direct acting pressure organs. 

B. RATCHET MECHANISM FOR PRESSURE ORGANS. 
I 319- 

Fluid Running Ratchet Trains. 

The pawls in a fluid ratchet train are the valves. They may 
be divided'into two great classes,t similar to those existing in 
ratchets of rigid materials, viz. 

Running Ratchets, or Lift Valves, and 

Stationarj' Ratchets, or Slide Valves. 

In the first class we have flap valves, also conical and spheri- 
cal valves, and in the second, the various iorms of cocks, cylin- 
drical and disc valves and flat slide valves. In both kinds of 
valves there exists an analogy to toothed and to friction rat- 
chet gearing, since by use of contracted openings the effect of 
friction is produced, and with full openings it is obviated. 
This gives a division which does not exist in the case of friction 
and toothed ratchet gearing. 

Viewed according to the preceding classification, piston- 
pumps, and piston machines are properly ratchet trains. | 
This idea does not seem to offer any practical diffrculties, since it 
can be made to include all the numerous variations without crea- 
ting more confusion than the former methods of classification. 
It is not practicable to distinguish between the devices acting 
by gravity and those acting by transfer of living force, since 
both are frequentl}' combined. 

The oldest devices are those using air, and the oldest piston 
is the membrane piston, (Fig. 949) in the form of a bag of skin 
used as a bellows. In this primitive device the earliest valve 
was the human thumb, and in the larger bellows the heel of 
the operator, these beiug followed at a later date by valves of 
leather.? The working part of the bag was next strengthened 
by a plate, (See Fig. 949 <r.) and developed into the common 
bellows, next followed the disc piston, a very early improve- 
ment II and later the plunger, from which the numerous modern 
forms have grown. The following examples will illustrate. 








Fig. 975. 

Fig. 975 a, is the [common lift and suction pump, a ratchet 
train similar to Fig. 749 ; a, is the pressure organ stream (cor- 
responding to the ratchet wheel a) b.^ the holding pawl in the 
form of a valve, f,, is the receiver or cylinder for the water and 
piston, fj, is a pawl-carrier in the form of the piston, b^, the 
other pawl, or lift valve. The water here overflows at the top 



* Used by Von Seydell in the Albert in 1S56 ; by Ruthven in the Water- 
■witch, 1S66, and recently in torpedo boats by Thornevcrofl. 

t See the author's Theoretical Kinematics, p. 459. ei seq. 

tThis treatment of the subject was first published by the author in Ber- 
liner Verhandlungen, in 1874, p. 228 et seq., but had previously been used in 
his lectures since !S66. 

§ Contrary to Wilkinson and Ewbank. the bellows shown in the Egyptian 
■wall paintiiigs have not flap valves, but the inlet opening is closed by the 
heel of the workman, and the bellows used to-day in India use the heel or 
thumb of the operator as an inlet valve. 

II See Belidor, Arch hydraulique, Paris 1739, II., p. 62. 



of the cylinder, and if it is to be lifted to a greater height the 
cylinder may be prolonged upward and the rod proportionately 
lengthened. If the rod is to be kept short, the form shown at 
b, is used. The top of the cylinder is closed and the rod 
brought out through a stuffing box, and the discharge tube 
only is prolonged. At c, is the so-called force pump with a 
disc piston, and at d, the same form with plunger. In these 
the discharge valve is in a separate chest. The water column a, 
is divided into two divisions a^ and (7,, the lower being impelled 
in the upstroke, and the latter on the down-stroke of the pis- 
ton. A blow or shock is produced at each stoppage of the 
motion of the water column and to reduce this action the speed 
of flow must be kept down, and also the shock cushioned by 
means of air vessels. At d, air vessels are shown both on the 
suction and force pipes. 

The preceding pumps are all single acting, discharging one 
cylinder of water for each complete double stroke of the piston. 
By cjdinder of water is here meant the product of the piston 
area by the length of stroke. 1| The space between valves and 
piston is not included, this being merely clearance or water space. 

The piston may be so constructed that it remains stationary 
and the cylinder slides upon it, this forming an inversion of the 
common form and possessing many applications. 

Fig. 976(7 is Muscheubrceck's pump (1762) for moderate lifts, 
b, is Donnadieu's pump for deep wells, especially adapted for 




M^ 



\-\ 

i 



ig 



Fig. 976. 

artesian wells.** This latter form possesses the peculiaritj- that 
cylinder and discharge pipe move, and the piston is statiouary 
while action is not changed. (See Fig. 749 ) At f, is Althaus 
so-called telescope pump, which does not differ from Fig. 975 a, 
except that the piston is longer and is operated by two side 
rods instead of a .single central one.tt The form at d, is a mod- 
ification of c, with external packing. 

In the pumps shown in Fig. 975 <7. b, and Fig. 9^6 a, the pis- 
ton rod plunges into the water on the downward stroke and 
hence acts as a piston, lifting water by its displacement. On 
the upward stroke the water flows into the space again, 
and so the volume of delivery is not altered but a slight portion 
of the delivery takes place on the down stroke. This action 
can be utilized, however, as was very early done in mine 
pumps, by increasing the diameter of the rod, or forming it in- 
to a plunger so as to cause the delivery to be divided equally 
between the two parts of the stroke. This form may be called 
a double delivery pump, or briefly a double pump, since it is 
practically two pumps, using the same set of .valves. Some 
examples follow. 

Fig- 977'') the plunger fj, is connected to the piston c,, the 
latter being twice the diameter of the former, this being the so- 
called "differential " pump. In b, two plungers are used, both 
valves being in separate chests XX At r, two telescopic pistons 
are used, this being by Rittinger, and well adapted for a mine 
pump. The form shown at d. has an auxiliary piston and cyl- 
inder parallel to the main cylinder, (designed by Trevethick in 



ly In small and medium sized pumps the loss of cylinder capacity diniin. 
ishes with the increase of speed. Experimental researches show 

at 27 to 40 strokes per minute 92 per cent. 
" 50 " " 95 

" 60 " " 98 

of the theoretical capacity (Konig, Pumps, lena, 1869). In very large pumps 
the momentum of the water shows an increase over the theoretical capac- 
ity: the pump in the Beryberg mine, i metre diameter giving 4 per cent, 
excess. See Portfenille John Cocquerill. 

** See Poillon, Traite th. et prat, des pompes. Paris, 1S85. Plate 27. 

tt See the design of the Spaniards, Barnfet, Viciauain & Poillon, Plates 33 
and ^4, and p. 193. 

U See Poillon, Plate 7, Saigun Waterworks. 

§§ See Ewbank's Hydraulics, New York, 1S70, p 280. 



224 



THE CONSTRUCTOR. 





Fig. 977- 

By making the suction valve also a moving piston, both the 
water columns may be kept in motion for both movements of 
the rod. This is a double acting ratchet mechanism (Fig. 750,) 
and hence also a double acting pump. 




Fig. 978 a, is a double acting pump with two opposing 
valved pistons, described by Fourneyron, but much older; this 
corresponds to the ratchet work of Fig. 750 a,. 

The pumps shown in Fig. 97S b, nnd r, are similar, the first 
by Stolz, the second by Amos & Smyth."* 

t 




Fig. 979. 

Fig. 979 a, is Vose's pump, in which the two pistons are 
placed parallel to each other. This corresponds to the Laga- 
rousse ratchet, Fig. 750 b. Similar double acting pumps may 
; be made with solid pistons, if it were 

desirable ; the form of Fig. 979 b, de- 
signed by the author, being an ex- 
ample, and others might readily be 
devised. t 

Fig. 979^, isDownton's pump. The 
three pistons (-3, c„, c^, keep the water 
in constant flow, which is further 
assisted by the air chamber. The foot 
valve 5,|, may be omitted if desired. 
, The annexed sketch of a pump by 
Lippold, (See Bach. Fire Engines, 
Stuttgart, 1883, p. 41,) is not double 
acting but contains practically on« 
piston split in two, and equivalent to 
one of half the area and same stroke, 
or two of the same area and half 
stroke. This is also the case with 




Fig. 980. 
Franklin's Double Pump, (See Konig, p. 55). 



By combining two complete fluid ratchet trains in such a 
manner that they have a common cylinder and piston, a form 
of pump is obtained which gives two full discharges for each 
cycle, and which may hence properly be called a double acting 
pump. 

b c 




Fig. 981. 



Fig. 981 (7 is a double-acting pump with disk piston, and Fig. 
981 b, the same form with a plunger. In both cases the suction 
pipe is at IV, and the discharge pipe at I. In double-acting 
pumps it is usually not convenient to put a valve in the piston ; 
this is, however, done in Fig. 981 c, in which we see two single- 
acting pumps combined in one. 

In Fig. 982 a, is shown Stone's Pump, % which is much used 




Fig. 982. 

for ships, as is also Downton's Pump. In this case there are 
four pistons, operating in two cylinders, the latter being placed 
one below the other on the same axis. The pistons Ci and c^ are 
connected by one rod and connected by the same crank k 1.3, 
and the other two pistons are, in like manner, connected and 
operated by the crank k 2.4, which is set opposite the other 
crank. The action may be more readily understood by examin- 
ing Fig. 9S2 b, which is similar to the preceding one, if we sup- 
pose the pistons r, and Cj to be held stationary and the other 
pair Cj, C3 driven by a single crank of double the length of arm 
of those shown. This will obviousl}' not alter the volume of 
delivery, and it will be evident that the lower pump is reallv a 
double-acting force pump and the upper one a single-acting lift 
pump, hence each revolution of the cranks will deliver three 
cylinders of water, two on the up stroke and one on the down 
stroke. In Stone's pump the pistons r, and c^ are so disposed 
that for each half revolution f cylinders of water are discharged, 
and in other respects the pump is a double-ratchet tram. Fig. 
982 c is Audemar's Pump. In this form two double pumps 
similar to Vose's Pump (Fig. 979 a) are combined to make a 
double-acting pump. ^ 



• See Theoretical Kinematics, p. 462. 



t See Poillon, Plate 29. 



(See Poillon, Plate 26. 
gSee Poillon, Plate 6, p. 93, 



THE CONSTRUCTOR. 



225 




C5) 




Fig. 983- 

Fig. 9S3 is Norton's so-called V shaped pump. In this device 
the pistons c^ and c\ form a single stationary piece, and the cyl- 
inder and valves b-^ and ^3 is 
the moving part. It will 
readily be seen how easily 
the lift pump may be made 
double-acting. 

A double-acting lift pump 
as used for a steam engine 
air pump, by Watt, is shown 
in Fig. 9Sz). This is practi- 
cally a combination of two 
different pumps. It has three 
valves, the foot valve 1^,, pis- 
ton valve 5[ and upper valve 
63. On the downward stroke 
the mixed air, water and va- 
por passes through the piston 
from the lower to the upper 
part of the cylinder, and on 
the up stroke this is dis- 
charged through b^ and a 
fresh cylinder full drawn in 
through b.,. This pump is 
double acting, since the pis- 
ton valve acts both in the up 
and down stroke. This works 
the same whether pumping 
liquid or gaseous fluids, the 
action being the same as if 
two valves only were used. 
The upper valve is required 
for other reasons, i. e. to con- 
trol the discharge, as for 
boiler feeding, etc. 
_ The preceding examples will serve to illustrate the applica- 
tion of fluid ratchet trains with running ratchets. It is impor- 
tant in all cases, and especially with the higher velocities, that 
provision should be made to have the valves close without 
shock, or in other words, that the engagement of the pawls 
should be quiet. This problem has already appeared in some 
forms of ratchet mechanism (see I 240) and here oflFers still 
greater difficulties, especially when heavy moving masses are to 
be controlled. The question is daily being considered in prac- 
tical problems of construction * and a great variety of valves 
has been designed. The present indications appear to be lead- 
ing toward the use of valves operated mechanically by the 
pump, instead of those operated by the fluid itself, but a final 
solution of this problem has not yet been reached. 

I 320. 

Fluid Ratchet Trains with Stationary Ratchets. 

As already shown in ? 255, it is necessary, in ratchet trains 
with locking teeth, to effect the engagement and disengagement 
of the pawls by some additional mechanism. This is also the 
case in those fluid ratchet trains which used stationary pawls, 
i. e., sliding valves. An example is found in the case of the 
simple single-acting air pump used in physical laboratories, which 
since its invention by Olto von Gerike t has been made with 
stationary pawls, and is shown in a crude form in Fig. 9S5. 

The "receiver" d', 
and its pipe connec- 
tion forms a negative 
reservoir, the pump 
a c d b^b^ a ratchet 
train for the propul- 
, us siou of the column 

P Or of air a. The suction 

riG. gs5. valve is at «„, and the 

discharge valve at 5,, both being in the form of stop cocks. 

The suction valve b., is operated by hand when the piston is 
drawn out, and when the end of the stroke is reached the valve 




.^1, which had previously been closed, is opened, and the first 
one closed, and the air expelled on the return stroke. A stop 
cock, b'i, is also placed close to the receiver. 

There is but little difficulty in apply- 
ing slide valves to single-acting pumps, 
and they are also readily arranged 
for double-acting cylinders. By exam- 
ining the arrangement of flap valves 
in the compress double-acting pump, 
Fig. 9S6, it will be seen that the valves 
b^ and b^ open and close simultaneously, 
and that the same is true of b„ and 63, 
and that the two actions alternate with 
each other. The operation of the valves 
is such that the four spaces /to /Fare 
connected alternately in the order /-// 
and /II-IV, and /-/// and II-IV. 
From this it will be seen that if sliding 
valves are used they may all be con- 
nected together, or united in the same 
construction. This may be done as 
shown in Fig. 987 a, which represents 
the so-called "four- way" cock. As here 
shown, all four of the passages are 
closed, this position corresponding to 
the end of the piston stroke. When 
the plug is turned 45°, as shown by the dotted lines, /and /// 
are connected, and also //and IV ; aud if it is turned the same 
amount in the other direction, /and // aud /// and /Fare 



'111 




l-i ' 




Fig. 9S7. 



connected. The portions bn and b^ may be omitted, as in Fig 
9S7 b, and the passages //, TVand ///brought closer together, 
as shown at c. From this form it will readily be seen bow the 
passage /can be converted into a mere delivery pipe, and the 
radius of curvature of the bearing surfaces, made of infinite 
length, giving the well-known slide valve. Fig. 9S7 b. In like 
manner other forms may be developed. It must not be forgot^ 
ten that this device really consists of four valves combined in 
one, and in fact recent forms of steam engines contain the four 
valves made separately, these often again being lift valves. 

A noteworthy peculiarity in the forms shown in Fig. gSy a and 
d must be considered. In both instances the valve overlaps 
the port on both sides, this being technically known as "lap." 
It is also apparent that the lap on the two sides of one port may 
differ, and that different laps may be used for different ports. 
By use of this expedient the opening and closing of the ports 
need not be simultaneous, but maj' occur successive!)'. 

From the preceding considerations the following propositions 
may be laid down ; the latter applying to all, and the former to 
nearly all, lift valves : 

The application of slide valves in all fluid ratchet trains de- 
pends upon two principles- 

1. The combination of several valves into one piece. 

2. The control of the time of action of these valves by means 
of the lap. 

I 




* See Fink.. " Konstruction der Kolben-und Zentrifugalpumpen," Berlin, 
1872; also Bach, "Konstruction der Feuerspritzen." 

i This name is spelled as given above in the earliest records, and not 
" Guericke," as is often given. 



Fig. 9S8. 

The application of a slide valve to a pump is shown in Fig. 
988 a. In this case /is the discharge outlet, and /Fthe suction 
connection. In such pumps it is necessary to provide some 
mechanism to operate the valve, and such mechanism is termed 
the " valve gear." This valve gear may be arranged in a great 
variety of ways. 

A simple form of gear is that shown in the figure, 988 a, in 
which an arm 6, attached to the piston rod, moves the valve by 
striking against tappets 5' and 5" on the valve stem. This 



226 



THE CONSTRUCTOR. 



arrangement is similar to the locking ratchet of I'ig. 753. It 
has the defect, however, of requiring the piston to move rapidly, 
or else the valve will not be carried past the middle position, 
and the purcp will stop. This defect can be met by using a trip 
gearing device such as shown in Figs. 74- and 743i to continue 
the condition of the valve when started by the impulse of the 
piston rod. 

A somewhat simpler method is that in which the reciprocating 
motion of the pump rod is used to revolve a shaft by means of 
a crauk, Fig. 9SS b, from which the valve may be operated by 
means of a return crank or eccentric. This arrangement is 
often used, especially for blowing engines, etc.* It will be 
apparent that a four-way cock device, Fig. 9S7, may be arranged 
so as to be operated by continuous revolution, instead of a 
reciprocating motion, and hence the eccentric may be omitted 
and a rotarj' valve device substituted. 

lu Fig. gSS b the crank and crank shaft are used merely for 
the purpose of actuating the valve gear. It is practicable, 
however, when a crank is once admitted, to use it still further 
as one of the parts of the pump, such as in chamber trains. 
Man}' such devices haae been proposed, f although but few of 
these have been put to practical use. The three following de- 
vices will illustrate. 






/////////M^/ /////////' 



Fig. 9S9. 

Fig. gSgfl is Pattison's pump, a form of chamber-crank train. 
7 he crank a here assumes the form of an eccentric, the rod b 
becomes a flat piston, the edges of which form a tight joint 
■with the ends of the cylindrical chamber d. In the position 
shown in the illustration the spaces //and /and III a.\\A IV 
are in communication. In the dotted positions /// is connected 
■with /, followed again by // and / and /// and IV. This trans 
fer of communication is produced by the action of the crank, 
and hence no other valves are necessary. 

The form shown at Fig. 989 b is made ■with an oscillating 
cylinder. The piece c, which plays an inconspicuous part in 
Fig. 989(7, is now used for the chamber, and its oscillating 
motion -with regard to b supplies the necessary valve action. 
Oscillating pumps are used in a variet}' of forms. 

Fig. gSgr is Beale's gas exhauster, made -with a so-called 
"sliding crauk" c, ■which acts at the same time as crank and 
piston. Without the use of special valves, the spaces // and 
///interchange with /and IVhy the revolution of d. Beale's 
exhauster is in successful and extensive operation in various 
gas works. 

In the examples cited and in the numerous modifications of 
them, it will be noticed that the checking or ratchet action of 
the liquid is invariably performed by slide valves. 

One of the objections to the use of slide valves for ordinary 
-water pumps is the wear upon the surfaces due to impurities 
iu the water. When the water is free from such objectionable 
impurities, it is to be considered whether slide valves might not 
be much more generally employed than has hitherto been the 
case. If this form of valve were given the benefit of practical 
study and experience, it ought to be possible to avoid the 
shocks due to concussion existing in pumps made with lift 
valves when operated at high speeds.J 

A great number of valve forms have been designed, 2 using 
combinations of single valves on the principle of the multiple 
ratchet (see ? 242), the action of the valves being assisted by 
weights, springs, etc., but these have not completelj' attained 
the desired end. II 



* See Zeitschrift Deutschcr Intjenieure, 1885, p. 929; also Herrmann's 
■\Veisbach's Mechanics, Vol. Ill . Part 2, p. 10S9. 

t See the author's "Theoretical Mechanics," in which over 90 chamber 
crank trains are described and analyzed. 

X Poillon refers to the fact that the automatic action of mechanically oper- 
ated slide valves enables hi^jh speeds to be obtained -^vith less noise than 
■when lift valves are used, but also notes the wear of the slide valves as an 
objection to their use. 

i See Riedler, Zeitschrift d. Deutscher Ingenienre, 1S85, p. 502 ei seg. 

\\ .See Bach, " Konstruktion der Feuerspritzeu," also in Zeitschrift d. 
Deutscher lug., 1886, p. 421. 



When the pump is used for pure water, as for drinking supply, 
the question of wear upon slide valves is not so important as 
with pressure pumps. A fair comparison can hardl}' be made, 
however, between pumps with slide valves and those with lift 
valves, as the former have been but little used and also not 
practically designed. 

It is a matter of surprise that when occasional applications of 
slide valves are made in pumping machinery, that such devices 
should be considered as something new. The difference be- 
tween the action of water and air is well known, and yet even 
with the slight weight of an air column the shock in blowing 
machines is most apparent. It can hardly be supposed that the 
other form would remain uninvestigated. 

The pumps shown in Fig. 9890 and c are commonly known 
as rotary pumps, which title is manifestly incorrect, since in 
form a there is an oscillating piston which does not rotate, 
while in form c, notwithstanding the rotary motion the action 
is similar to form a. Other so-called rotary pumps have been 
devised with curved piston action, some of these being as early 
as the 17th century. Iu some designs a radial slide acts in the 
pump case as a ratchet, and is drawn in and out by a cam of 
appropriaely curved profile. A large number of rotary pumps 
have been made on this principle, many of which will be found 
in Foillou's treatise. These pumps are usually made with 
metallic packing onl}', and are used in Italy and France for 
pumping wine and olive oil ; they are also adapted for brewery 
pumps. 

The undeniable predilection in favor of rotary pumps on the 
ratchet train principle is worthy of consideration. It is claimed 
that they have a higher efficiency, but this remains to be estab- 
lished ; also the rotary motion gives a continuous uniform motion 
to the wat;r column, but this is equally accomplished b}' the 
forms shown in Figs. 9S2 and 9S9. This uniform flow can only be 
approximately attained, as must be the case from the nature of 
the mechanism. The principle is that of a ratchet train which 
is intermittent in principle, and hence differs from a continuous 
running movement. The idea that such pumps give a continu- 
ous and uniform discharge is due to the fact that the column of 
water is operated directly from the part which is driven con- 
tinuously, but this by no means follows. This combination of 
a continuous running motion, with an intermittent ratchet 
action which is not apparent to the ej'e, will be sho^wu in other 
cases hereafter. 

i 321- 
Escapements for Pressure Org.\ns. 

Ratchet trains found with pressure organs also include 
escapements as completely as is the case with the preceding 
forms of rigid ratchet mechanism. The ratchet of ^ 258, shown 
again in Fig. 990 may be considered as an escapement if we 
assume the checking of a by 5 to be uniformly opened and 
closed. 

If now, in Fig. 991, the checked member a is made a pressure 
organ, such as water, in communication at //with a pressure 
reservoir, or with a negative reservoir at T, or both, the 
regular lifting aud closing of the valve b produces an escape- 




FiG. 990. 



ment acting in a similar manner to Fig. 990. By means of 
such a device the pressure organ a can be constrained in per- 
forming mechanical work. The range of such an escapement 
is not determined by the teeth of a wheel, but on the contrary, 
is similar to a friction ratchet, and can be varied at will. 

The applications of escapements wiih fluids are in principle 
the same as those formed of rigid bodies, but in practice their 
nature is very different. We have already distinguished between 
■watch escapements and power escapements, and iu the present 
instance the power escapements are by far the most important. 
For this reason the latter will be considered first. Unperiodical 
escapements are shown in the simple form of Fig. 991, in which 
the time of releasing and checking is regulated by hand ; a 
form ver3' seldom found in rigid escapements. Periodical 
forms, similar to watch escapements, are used with pressure 
organs for measurement, but not for measurement of time, but 



THE CONSTRUCTOR. 



227 



of volume. To these we tuaj' add the adjustable escapements 
on the principle of those described in \ 259, and we have the 
following classification : 

a. Unperiodical Power escapements. 

b. Periodical Power escapements. 

c. Adjustable Power escapements. 

d. Escapements for measurements of volume. 

A. UNPERIODIC POWER ESCAPEMENTS FOR PRESSURE 
ORGANS ■ 



Fi^uiD Escapements for Transportation. 

One of the simplest practical applications of the principle of 
Fig. 991 is Felbingex's Postal Tube, shown in diagram in Fig. 
992. The line tube d is connected with a reservoir of compressed 



reversal of iSo°. This may be accomplished either by the use 
of tension organs or pressure organs. 




n .^;^c- 



T 



Fig. 992. 

air at /I, and at 7" with a similar negative reservoir. At 6 is a 
sliding pawl, here shown open ; the piston, or carrier c, in the 
form of a leather box containing letters, telegrams, etc., being 
driven through the tube. A valve b' enables the end of the 
tube to be thrown into communication with a second negative 
reservoir, and this mechanism can be arranged at both ends of 
the line so that the tube can be used for transmission in either 
direction. Such postal pneumatic tube:- are well known and 
widely used.* 

An atmospheric escapement operated by a negative reservoir 
is found in the so-called "atmospheric railway," invented by 
Pinkus in 1834, and put into practical operation somewhat later 
in England by Clegg and Samuda. This was operated on the 
Kingston-Dalby road with a vacuum of '/e atmosphere in the 
exhausted receiver, but it is no longer in operation. 

When an escapement is intended to control the back and 
forth movement of a piston in the same path, the single valve 
shown in Fig. 991 is not sufficient, but at least a second must be 
used, as is already indicated in Fig. 992. One of the most prac- 
tical of all fluid escapements is found in the lock used on canals 
and shown in diagram in Fig. 993. 



-b2- 




bi 



■l//l//H///////'l» ' • ///////■•/ 



'/4 ^""'"" % ■■ -. 






T- 



bj 



bi 



Fig. 99 > 



The canal is open on the upper side (see Fig. 945 b and c) ; 
the valves b^ and b., are of the running ratchet form, and are in 
reality double gates. Smaller by-pass valves b^' and b,' are used 
in order to enable the inlet and outlet of the water to be started 
gradually. The boat c forms the piston, and when the motion is 
upward, b^ is the escapement valve, and when downward, b.^ is 
used. 

The above canal lock device, while extremely useful, pos- 
sesses a very low efficiency, since it not only uses a volume of 
water equal to the displacement of the boat plus the necessary 
clearance, but also discharges the whole lock chamber of water 
each time it is used. Later devices have been made for the 
same purpose, involving a less waste of water. If it is arranged 
for the service to be doubled by making two lifts adjacent to 
each other, it is evident that the descending boat can counter- 
balance an ascending one of the same weight, the only require- 
ment being that there must be some connecting mechanism in- 
volving the overcoming an additional resistance, and capable of a 




Fig. 994. 

Fig- 994 shows a double canal lift constructed by Green tax 
the Grand Western Canal in England in 1840, the connecting 
mechanism being tension organs in the form of chains. The 
boats are carried in tanks c^ c.,, the ends of which are closed by 
valves or gates 6, and b.,. and similar gates b/ and b./ also close 
the ends of the canal sections. A small addition to the weight 
on the descending side is sufficient to raise the other tank f " 




Fig. 995. 

The substitution of a pressure organ for the chain was first 
made by Mr. Edwin Clark on the Mersey Canal in 1875, in the 
form of a hydraulic lever, as shown in Fig. 995. This shows 
clearly the equivalence of the cord or chain and pulley and the 
water lever, already referred to in i 311. The tanks rj and c^ 
are carried on plungers 3 feet in diameter, and are 75 feet long 
and I5j4 feet wide. A head of 6" of water is sufficient to over- 
come the resistance of motion, and a lift of 50 feet is effected 
in three minutes.J Smaller installations have been made by 
Clark and by Stanfield, and other large ones at the La Louviere 
Canal in Belgium, and the Neufosse Canal at Les Fontinettes, 
in France. The lifts are 43 ft. and 50 ft. respectively, and the 
plunger diameters 6|< feet. The loss of water with these lifts is 
only about i^Jj of the quantity used by common locks of the 
same capacity. ^ 

The preceding escapement devices are made for open canals, 
but escapements may also be constructed with closed tube con- 
nections. This latter type includes numerous hydraulic eleva- 
tors for lifting burdens of all kinds. 

An example of a direct-acting hydraulic elevator is given in 
Fig. 996. The two valves are combined in one cock. The water 
under pressure enters at //, and the discharge against the 
atmospheric pressure is at A. The weight of the plunger is 
counterbalanced by two counterweights G with chains and 



•In 
miles. 



' the length of the postal pneumatic tubes in Berlin was over 26 



t See Weisbach-Hermann, Vol. III., Part 2, p. 633. 

X See Duer Trans. Inst. C. E., 1876 ; Colyer, Hydraulic Machinery, London, 
Spon., iSSr, p. 17 ; also Robinson, Hydraulic Machinery, London Griffin & 
Co., 1887, p. 64. 

g See Colyer, p. 29 ; Robinson, p. 69 : also Zentralbl. der pr. Bauverwaltuug, 
1SS2, p. 395 ; Hensch, Schiftshebung in Frankreich ; also Scheiufil, Kanal 
and Hafenwerkzeuge in Frankreich und England, Wien, Ceroid, 1S82, p. 15; 
also Ernst, Hebezeuge, Berlin, Springer, 1S33, p. 630. In Green's lift the 
loaded boats descended and the empty ones ascended, hence an excess of 
water was raised, which was permitted to overflow. These lifts enable much 
greater differences of level to be overcome than do the ordinarj' locks, and 
make it practical to use long stretches of canal and make an entire lift at 
one operation. It may be here noted that pneumatic lifts for canals were 
designed in 1863 by the Swiss engineer. Sevier. 



228 



THE CONSTRUCTOR. 



pulleys, and the plunger operates the valve automatically by 
means of the rod b' , when the highest 
position is attained. This form of lift has 
been much used, sometimes of very large 
dimensions. The great passenger elevator 
of the Hamilton St. Station of the Mersey 
Tunnel has a plunger iS" in diameter, 
with a lift of Syjj feet, the car holding 50 
passengers.* 

A practical objection to direct-acting 
lifts of this form lies in the heavy counter- 
weights required, and also in the depth to 
which the cylinder must be sunk. A 
different form has therefore been designed 
in which a piston travel of moderate length 
is multiplied by use of a tension organ 
system, such devices being extensively 
used for passenger elevators, notably by 
the Otis Elevator Company. 

Hydraulic cranes are also forms of high 
pressure escapements, first designed' by 
Armstrong, and since used by many others, 
especially in connection with Bessemer 
Steel plant, in which hydraulic cranes 
have proved most valuable. 

Fig- 997 shows the mechanism of a 
hydraulic crane by Armstrong. The piston 
is double acting, and there are four valves 
/^i, b,, b^, b^, of the type shown in Fig. 9S6, 
the external connections also being neces' 
sary in order to complete the escapement. 
The high pressure water enters at //, and 
passes through the pipe /, and is discharged to the atmosphere 
at IF. The rod fj is made of half the area of the piston e^ 





Fig. 997. 

(compare Fig. 946 e). When bi and b^ are open, as in the illus- 
tration, the forward stroke is made with one-half the full force ; 
when 5j and 5^ are open, the forward stroke is made with full 
force. By opening b., and b.^, the return stroke is made by the 
pull of the load upon the chain. At b' is a safety valve which 
comes into action should the load descend too rapidly, by the 
opening of b^ alone.* 

? 323- 
Hydraoi,ic Tooi^. 

Hydraulic escapements, similar to those used for lifting loads 
are also applicable to machine tools. Among these may be 
noted the devices of Tweddell, for riveting, punching, bending, 
etc. (see ? 54). 

Figs. 998 and 999 show the arrangement of Tweddell's rivet- 
ing machine ; d is the piston, bi, b.^ the valves, one of which 
connects with the pressure reservoir at M, and the other with 
the atmosphere at ^. When b^ is opened by the lever e, the 
hydraulic pressure enters above the piston d, and the stroke is 
made. The return stroke is effected by means of the auxiliary 
piston rfj, which is fast to d, and under which the water pres- 
sure is acting at all times. Clusitig b-^, and opening b^, enables 
this to act and lift the main piston. This gives practically a 
hydraulic Jever of unequal arms, the shorter arm always being 
loaded with H, and the load on the longer arm varying between 
H and A. The lever mechanism d', d'\ d"', controls the 
length of stroke of the die, by means of the tappets d" and 
d'", which are connected with the lever e. This is also used 
on the lift of Fig. 996, and shows the complete escapement. 
The arrangement of valves is shown in detail in Fig. 999.5: 




Fig. 998. Fig. 999. 

The preceding apparatus resembles the hydraulic press. It is 
in fact quite different, being a genuine ratchet train, capable of all 
the modifications of such mechanisms as to speed, distance, and 
arrangement. On account of these points the applications of pres- 
sure organ escapements are becoming rapidly more important. 

§ 324- 
Pressure Escapements for IMoving Liquids- 

The use of unperiodic pressure escapements for moving 
liquids in machine construction has 
been practiced from an early period, 
and at the present time improved de- 
vices for this purpose are much used. 

An almost forgotten device of this 
kind is Briudley's boiler feeding ap- 
paratus, Fig. 1000, this being based 
upon the principles already given in 
Fig. 991. 

The necessary opening of the valve b 
is made by the float c, and the closing 
by the counterweight c, (compare Fig. 
950). This apparatus was first applied 
to Watt's boilers, the feeding of the 
boilers of Newcomen's engines being 
effected by a cock operated by the 
attendant. 

Fig. lool is Kirchweger's steam trap 
for the removal of water of condensa- 
tion. The escape valve b is opened by 
the float c, which, in this instance, is 
open at the top, so that the water flows 
over the rim until it sinks, and thus 
opens the valve, This valve motion is 
in itself a ratchet train, checked and 
released by the action of the float. 
When the valve is opened the water in 
the float is forced out by the pressure of 
the steam. § 

The slow moving float device, as in 
Fig. 1000, has also been advantageously 
for operating steam traps, by 




used 



Fig. 1 000. 



* See Robinson. 

t See Weisbach-Herrman, III., 2, p. 240 ; Colyer, p. 11 ; Robinson, p. 52. 

X For fuller descriptions of Tweddell's machine see : Proc. Inst. C. E. 
LXXIII., 1SS3, p. 64 ; EngiHeer, July, 1S85, p. 88 ; August, p. iii ; Revue Indus- 
trielle, 1S84, p. 5: 1885, p. 493; Mechanics, 1885, p. 272; also Rol»inson, as 
above, and Zeitschr. Deutscher Ing., 1886, p. 452. 



Tulpin, of Rouen ; Handrick, of 
Buckau ; Puschel, of Dresden; 
Dehne, of Halle, and others. 
Similar escapements have been 
designed to separate air from 
steam, or air from water, as in 
the devices of Andral, Kuhl- 
mann, Klein and others. || 

Other examples of escape- 
ments of this kind are found in 
the so-called Montejus, used for 
elevating syrup in sugar refiner- 
ies, in the return traps of .'-team 
heating systems, and in various 
other forms of boiler feeders, 
such as those of Cohnfeld. Rit- 
ter & ISIayhew, and others. ^ 

§ This form of trap is made in many varieties, tlte one shown being by 
IvOSeuhausen, of Dusseldorf. A similar one by MacDongal is much used in 
Bngland, and a feed pump on this principle is made by Korting in Hanover ; 
German Patent No. 3'^-, 332. 

(1 For illustrations of these devices s-ee SchoH's Fiihrer des Maschinisten, 
10 Bdit., p. 493- H See SchoU, p. 235. 




Fig. iooi. 



THE CONSTRUCTOR. 



229 



B. PERIODICAL PRESSURE ESCAPEMENTS. 
k. 325- 

PoMPiNG Machinery. 

Periodical fluid escapement trains have a wider application 
than unperiodical trains, since it is practicable, as already shown, 
to use a fluid ratchet train to operate the valves in a simple 
manner. This makes it possible to produce the opening and 
closing of the valves in a periodical succession mechanically, 
instead of by the fluid column. In this construction the fluid 
■column may therefore drive the piston, instead of being driven 
by it. This idea seems very simple, and yet pumps had been 
known for two thousand years, and had occupied the inventive 
energy of the preceding centuries before the siinplest forms of 
the modern steam engine were devised. It is therefore all the 
more important in the study of machine design to investigate 
the fundamental principles involved. 

It is impossible, in the limited space which can here be given, 
to go into this subject iu its entirety ; the arrangement of the 
valve gear of the Newcomen engine with tumbling bob gear, is 
an instructive example. 

In Fig. 1002 is shown Belidor's single acting water pressure 
engine." 



in communication with the discharge, and since b.^ is larger than 
5,, the pressure between them moves them into the position 
b^' b./. This puts the main cylinder in communication with the 
disciiarge, and the piston sinks by the weight of the load upon 
it. At the close of the stroke the tappet 6 moves the arm c/ 
into the position c-^ again, and places the auxiliary valve in the 
first position and a new stroke is made.i 

This machine constitutes an escapement of the second order, 
since the small and large escapements alternately release each 
other ; the lever device 5-6-Ci forms a third mechanism, so that 
the machine, as a whole, is of the third order. 




Fig. 1002. 

In the cylinder a' is a piston ; rt, is the entrance of the water, 
«2 the discharge outlet. The valves *, and A, are united in a 
three-way cock (see Fig. 987). This valve is operated from the 
piston rod <: by a tumbling-bob gear (see Fig. 742). The tum- 
bling lever E t\ e.„ weighted at £, is connected with the piston 
rod at c,, and moves about its axis independently of the lever/. 
When the end of the piston stroke is nearly reached, the lever 
E passes the middle point, and tips over, when the arm/, strikes 
the lever/ and carries it to the position /, moving the lever of 
the three-way cock from b to b'. The arm f, is behind E. The 
return stroke of the piston moves the arm e.^ of the tumbling 
gear towards the right, and as the end of the stroke is reached, 
the tumbling bob is again tripped, and the three-way cock 
moved again into the position b. A cord secured at the ends to the 
points ^3 and c^, and fastened to E, limits the travel of the latter. 
The piston rod is connected directly to the pump to be operated. t 
It will be observed that this machine is a ratchet train of the 
second order, the piston and valve forming an escapement, and 
the valve gear a releasing ratchet train each operating the other. 

Fig. 1003 is the single 
acting water pressure en- 
gine of Reichenbach. In- 
stead of using a tumbling 
bob gear to operate the 
valve, Reichenbach uses 
a second water escape- 
ment, operating the valve 
by a piston, the valve 
being itself a piston valve. 
The double piston valve 
^3 bi of the secotid escape- 
ment is operated by the 
main piston rod, the tap- 
pets 5 and 6 striking the 
lever fj as each end of the 
stroke is reached. The 
water under pressure en- 
ters at <7| and is discharged 
at a.,. The tappet 5 moves 
the auxiliary valve into 
the position b/ 5/, which 
places the space above b^ 

* Belidor, Architecture hydraulique. Paris, 1739. Vol. II., p. 23S. 

t The atiove described machine, designed by Belidor in 1737, for the water 
■works at the bridge of Notre Dame, does not appear to be altogether ^jracti* 
cable. It has been here given on account of the valve motion, which is his- 
torically interesting and doubtless good, and has been re-invented several 
times since. It was not new in 1737, having been in Newcomen's steam 
engine, as was already known to Belidor, since it is described by him iu the 
same volume of his treatise. 




Fig. 1003. 




Fig. 1004. 

Fig. 1004 shows the double acting water pressure engine of Roux. § 
The double action is obtained by combining the four valves in one, 
and by communicating the admission and discharge alternately 
with both sides of the piston. In this case the lever connection 
Ci is replaced by an escapement. The small pistons b./ b/ are 
acted on at the outer ends by the pressure water through the 
small passages k./, k/. This gives an escapement of the third 
order. The cup-shaped ends c.^, r.,, of the main piston c form 
the pump plungers. This machine should operate satisfactorily. 
It is readily apparent that the piston steam engine may also 
be considered as an escapement. The valve gears differ from 
the preceding forms only on account of the conditions of ex- 
pansion and condensation. These are reducible to a limited 
number of simple cases. 




Fig. 1005. 
Fig. 1005 is a single acting high pressure engine. The steam 



t See Weisbach-Herrnian, II., 2, p. 536; a'™ Ruhlmann, allgem Masch- 
inen Chre., l.,p 34S . . , ^■ 

8 See Revue Industrielle, 1S84, p, 114- Built by Crozet et Cie. 



2^0 



THE CONSTRUCTOR. 



enters at <7|, and the discharge to the atmosphere is at a^. The 
opeuing of the valve b^ permits the steam to enter, forcing the 
piston c down, and raising the weight G. The valves b^ and b.^ 
are operated by a ratchet train released by the tappets 5 and 6 
on a rod moved by the main piston. The pawls are double- 
acting, and are of the form shown in Fig. 67 r. When c reaches 
the bottom of the cylinder the tappet 5 releases the ratchet 7, 
and closes the valve b^ by means of the connectionsy, e^ The 
release of 7 opens the valve b., by means of the connections e„f^, 
and permits the escape of the steam from below the piston. 
This equalizes the pressure above and below the piston, from 
which the valve b., is called the equalizing valve. The upward 
stroke of the piston causes the tappet 6 to reverse the ratchet 7 
and operate the levers e-^f^, closing the equalizing valve and its 
connections. 

The device differs from the preceding in that the principal 
escapement a b^b„cd changes in character with the stroke. 
The two ratchet trains can be seen in principle in the double 
acting tumbling gear of Fig. 1002. The mechanism, when lift- 
ing the valve, is of the third order, and when closing, of the 
second order. The gear as shown is Farey's ; Fig. 779 shows 
this principle in a rigid escapement train, the corresponding 
form in single-acting train is the chronometer escapement, Fig. 

If the engine is a condensing one, a condenser valve dj is 
added, this being opened by the closing of b.^, as is also a jet 
valve in the condenser. When the steam is to be expanded, the 
lever e-^ is so arranged, the closing of (5, is produced earlier (see 
the smaller diagram) by the position of the tappet 5, and the 
corresponding counterweight lifted. This only operates the 
ratchet 7, and f,_ is released by a second train 8, which is 
effected by the tappet rod or by the so-called cataract K, 
released by a tappet 9, see \ 260. 

The condenser is a negative reservoir, and was the principal in- 
vention of Watt. It involves the use of two fluid ratchet trains ; 
the air pump, and the cold water pump, and also usually in- 
cludes a boiler feed pump. The entire engine is composed of a 
collection of ratchet trains. 

Steam pumping engines are by no means always made with 
lift valves, and a great number of more recent designs are made 
with slide valves (see Fig. 9S7). Ritteuger has applied slide 
valves successfully to single-acting engines, and they are espe- 
cially applicable to double-acting non-rotative engines. In the 
last decade especially' have valve motion for steam pumps with 
slide val.ves been multiplied, and some illustrations are here 
given. 

Fig. 1006 is Tangye's direct-acting steam pump. The steam 




iill ^ \M. 

Fig. 1006. 



entrance is at /, and the exhaust at IV. The slide valve b is 
the so-called E form, combining the four valves of Fig. 9S6 in 
one ; b^ and ij are the auxiliary pistons to move the valve, and 
form part of an escapement ot which the valves b" and b'" are 
operated by the mam piston c at each end of its stroke. The 
latter valves communicate with the cylinder posts //and ///. 
When b"' is lifted by the piston, the space R is in communica- 
tion with the exhaust, and the pressure in Z throws the valve 
over, equilibrium being soon after established through the aper- 
ture k,,. The reverse action occurs on the return stroke. This 
is a steam escapement of the second order, with an independent 
starting lever, the whole forming a combination of the third 
order. This has been much used by Tangye for steam pumps. 

Fig. 1007 shows the valve motion of the Blake pump, which 
is very extensively used in the United States. In this case there 
is a movable seat b^ under the valve b, the opeuing through the 
seat always being in communication with the posts //, ///, 11^, 
although ba is moved a short distance at each end of the stroke 
by tappets on the piston rod. In the position of the parts 
shown the steam entering at /will pass through ///and move 



the main piston to *he left, as indicated by the arrow. Just 
before the end of the stroke is reached the seat b„ is moved as 
much to the left of the centre as it now stands to the right. In 




1 f /f 



n_ 

// / / /w/m 



^M^ 



Fig. 1007. 

the seat b^, as shown in the figure to the right, there are addi- 
tional valves formed, /?,, ft, ft, which act to operate the auxil- 
iary pistons 1^2, (Jj, under which latter the small steam passages 
can be partly seen. When i')^ is moved to the left, a small post 
is uncovered by ft, and live steam enters the cylinder L behind 
A^, while at the same time ft connects R with the exhaust. This 
causes 5,, b, b^, to move to the right and reverses the pump. 
The reverse action takes place at the other end of the stroke, 
the whole forming a combination of the third order. 




■///////////////A 



n '/>/ ! Y //>. in , 

)7///////////////////////////////My////^^^^^^ 



Fig. 1008. 

Fig. looS shows the valve gear of Deane's steam pump, which has 
also been extensively used. The main valve is moved by means 
of auxiliary pistons, as in the preceding instance. The auxil- 
iary pistons are controlled by a separate valve b' , which itself 
is operated by lever connections with the main piston rod. 
This combination b' , b-, b, forms again a mechanism of the third 
order.* 

If the last three devices described are compared with the 
Reichenbach water-pressure engine, it will be seen that the 
fundamental principle is the same in all. The constructive 
arrangements which may be adopted are clearly shown in the 
precedi'ug examples, which may be modified in a variety of 
ways. Among other widely used arrangements, that of Knowdes 
maj' also be mentioned ; in it the action of the auxiliary pistons 
is controlled bj' a slight twisting motion given to the valve stem. 



..^.^^oJm 




Fig. 1009. 

Fig. 1009 shows Pickering's steam pump t In this design the 
main piston c acts also as the valve for the auxiliary pistons 
b,, ^3, so that the spaces R and L are placed alternately in com- 



* See .A.m. :\Iachinist, Feb. 17, 1SS3, p. 4 
Dow, see ;Mining and Scientific Press, 18 
t See Poillou. 



For an excellent steam pump by 
i, March, p. i6g, and May, p. 313. 



THE CONSTRUCTOR. 



231 



munication with / and IV. The whole forms a steam escape- 
ment of the second order. 

Fig. loio shows Harlow's valve gear, also used for pumping 
machinery.* This is also a steam escapement of the second 
order, similar to the preceding. The valve action for the 
auxiliary pistons is formed in a prolongation of the piston rod, 
the grooves c^ and €.-,_ placing the spaces R and L alternately in 
communication with IV. 




59* 



Fig. ioio. 



By comparing the preceding designs with the water pressure 
engine of Roux, Fig. 100.J, the similarity will be apparent. All 
the examples given show the fundamental relation existing be- 
tween these devices and the mechanical escapements of watch 
movements. The escape wheel is replaced by the fluid column ; 
the anchor, by the valve ; the vibrating member, whether pen- 
dulum or balance wheel has. here not a free movement but a 
determinate one against an external resistance. Similar 
arrangements include steam hammers, also hammers and rock 
drills, usually driven bj' compressed air, these latter consisting 
of mechanism of the second, rather than the third order. An 
example will serve to illustrate the general arrangement of such 
devices. 




Fig. ioii. 



Fig. loii shows the arrangement of Githen's rock drill. f 

The curved valve 4, is operated by the action of the curved 

outline formed in the piston c. The middle position of the 

valve is a dead point, but this is overcome by the momentum 

of the heavy piston. 



The devices of the third order are capable of a very import- 
ant modification which can be considered by examining for in- 
stance the Deane gear. Fig. looS, or either of the two preceding 
it. An inspection will show that it is entirely practicable to 
use the auxiliary piston to operate a pump cylinder, as indepen- 
dently of that operated by the main piston d. It is only neces- 
sary to make it larger in diameter and of proper length of 
stroke ; and there is nothing to prevent making it of the same 
diameter and stroke as the main piston. 

The valve of each cylinder will then be operated by lever 
mechanism connected to the rod of the other piston. This 
arrangement involves the replacing of the E valve by the com- 
mon D valve, which is not important, but is nevertheless an 
advantage. The two escapements are conveniently placed side 
by side for constructive reasons, and the double arrangement is 
known as a "duplex" machine, this term being given to two 
combined cylinders, of which the valve of each is operated by 
the piston movement of the other. This type is now frequently 
met, having been made for small apparatus very early, in 
France by Mazellire and yet earlier, in 1S59, ii^ the United 
States by Worthingtou. 




Fig. 1012. 

Fig. 1012 shows a duplex pump by Mazelline.j The illustra- 
tion shows one piston c^, at mid-stroke with its valve ij, at the 
end of its travel, and connected to the rod of the other cylinder 
by the lever e„. 

The work is divided into two portions which is provided for 
by the doubling of the parts. If the two piston escapements 
(cylinders, pistons, valves, steam, etc.) are indicated bj- [i] and 
[3], and the valve movements by [2] and [4] the action will be 
as shown in the following lines, 




whence we have 
and 



[1] M [3] 
[3] [4] [I], 



both being of the second order. 




* See Engineering and Mining Journal, Oct., 1884, p. 23 
t See Eng. and Mining Journal. March, 18S7, p. 107. Also Halsey's rock 
drill, Trans. A. S. M. E-, 1S84-5, p. 71. 



Fig. 1013. 

Fig. 1013 shows a perspective view of Worthington's Duplex 
Pump, the arrangement of which is apparent from inspection. 
The duplex steam cylinders are at the right, and the double 
acting pump cylinders on the left. 

The advantages obtained by using this form of pumping 
machine practically outweigh the objections which might be 
made against the duplication of pa.rts. In double acting pumps 
of the forms shown in Figs. 1006 to loio, the motion of the 
water columns is interrupted, at low speeds, at each reversal of 



tSee Poillon, Plate I-X 



232 



THE CONSTRUCTOR. 



the piston, while with the duplex pump the discharge is practi- 
cally continuous, because each cylinder begins its stroke just 
before the other comes to rest. 

An objection to all the other forms of direct acting pumps 
already described lies in the fact that to obtain uniform pump- 
ing action it is necessary to carry the initial steam pressure for 
the entire stroke of the piston, or in other words, the best action 
of the water end is obtained by means of the least economical 
action of the steam cylinders. 

This defect was overcome in the earlier pumping engines, 
such as the Cornish mine engines, by using the steam to lift 
heavy weights, pump rods, etc., the living force of the mass 
permitting an early cut-off and high expansion, and the uni- 
form descent of the weight being used to force the water. By 
this method the Cornish engines attained a high degree of 
economy. This method being single acting, caused the entire 
column of water to come to rest during the time required for 
the up stroke of the pump rod, and hence the Cornish type of 
pumping engine gives a most economical action of the steam at 
the expense of a defective action of the pumps. 

In the larger sizes of Worthington pumping engines the ex- 
pansion of the steam has been for a long time effected by using 
compound cylinders, and excellent results attained in steam 
economy. The efficiency, however, was by no means so high 
as was desired. In iSS6 the so-called Worthington equalizer 
was introduced with a view of enabling the desired high duty 
to be attained. 





Fig. 1014. 

This device, shown in Fig. 1014, is a ratchet train of the tumb- 
ling type, similar to that shown in Fig. 743, the springs being 
replaced by water pressure from a high pressure air cham- 
ber.* The air chamber forms a periodical storage reservoir. 

The plungers/', f, are attached to a cross-head connected to 
the prolonged piston rod, and the cylinders are carried on 
trunnions 7, 7. During the first half of the stroke the plungers 
are forced into the cylinders the latter swinging about the cen- 
tres 7, 7 ; and during the second half they are forced out by the 
action of the stored energy .t 

The resistance and assistance which the pistons /give to the 
steam piston is shown by a curve of the form of Fig. 1014 b. as 
has also been obtained by the indicator. 





Fig. 1015. 

If in Fig. 1015 a, we make P equal the component on each 
portion of the pressure O on the main piston rod, we have : 

_ D • ^ 2 Ptan/3 



in which 



This gives 



tan p ■■ 



•v^i -t- tan /3' 

A' 

T 



P 



Q-- 



^^(ff 



* A.n equalizer of this t3'pe was patented in Germany, by the Berlin 
Anhalt Works in 18S5. 

tin kinematic notation this action is expressed by CP i-CP-l), as shown 
l)j' ^. See Theoretical Kinematics, pp. 322, 325. 



or if we make Q the ordinate,]', of the desired curve : 
__2jPa-_ 

and substituting c for 2 Pa- 

we have ., ^ 



c s/^i 4. tji 



(317) 



which equation is readily expressed graphically. 

If this curve is drawn upon the rectangle which represents 
the resistance of the water, as in Fig. 1015 b, we get the actual 
resistance curve/"^ //, and this resembles closely the expansion 
line for a high degree of expansion, or in other words, the im- 
pelling force and the resistance are practically made equal to 
each other througho"t the stroke. The dotted curve a b c d e, 
is that of an actual indicator diagram, j This shows that with 
the Worthington high duty pumping engine the most efi&cient 
action of the steam is obtained at the same time as the best 
action of the water end. < 




Fig. 1016. 

Fig. iot6 shows a longitudinal section of a Worthington high 
duty pumping engine. The equalizing cylinders and their air 
chamber are seen on the right ; the dotted lines e,^ show the rod 
of the second cylinder, which operates the valve b^. 

As it has already been seen that many forms of the third or- 
der can be reduced to the second order, it may be inquired as to 
the possibility of obtaining a pumping mechanism of the first 
order. This has already been accomplished by uniting the 
steam escapement with a water ratchet train. The device is the 
Hall Pulsometer, shown in diagram in Fig- 1017. 

The steam enters at a, at b, is the anchor shaped pawl, and d, 
is the vessel corresponding to the frame- 
work of a rigid escapement, (compare 
Fig. 775). If the vessel d is closed as 
shown by the dotted lines and a volume 
of water c, included, we obtain an action 
of the first order. The efficiency is very 
low ; about % to ^ that of a piston 
pump, but the simplicity and conven- 
ience is so great that this may often be 
neglected. 

Another escapement of the first order 
is Montgolfier's hydraulic ram, which is 
a w-ater checking-ratchet train, the effi- 
ciency of which is low. A more recent 
device is the application of a water 
ratchet train to drive a pneumatic rat- 
chet train, first used on a large scale by 
Sommeillier in the construction of the 
Mont. Ceuis tunnel, and by means of 
which the efficiency was brought up 
nearly to 50 per cent.|] Pearsall has re- 
cently improved the hj'draulic ram and 
raised its efficiency to nearly So per 
cent., either for water or for air, but this 




FiG. 1017. 



\ See Mair, Experiments on a direct-acting steam pump. Proc. Inst. C. E. 
London, 18S6. 

j! The Worthington equalizer accomplishes an end sought by designers or 
steam pumps for the past 200 j^ears, for since Papins first machines in Cas- 
sel {[690) the desired aim has been to combine the action of a variable elastic 
driving medium, and a uniform, non-elastic resistance. 

II See the autlior's paper, Ueber die Durchbohrung des Wont-Cenis. 
Schweiz. polyt. Zeitschr, 1S57, p. 147. 



THE CONSTRUCTOR. 



^33 



has been done by the introduction of a valve gear, making it a 
device of the second order.* 

§326. 

Fi:,uiD Transmission at Long Distance. 

When the motive power is intended to operate the piston of 
a pump situated at a distance, some connecting mechanism 
must be interposed between the two cjdinders. Formerly this 
was accomplished by using long rod connections; instead of 
this a pressure-organ transmission may be employed. When 
water is used as the medium for transmission, this may be 
termed a " water rod " connection. This is used in connection 
with water levers (see \ 311). 






§327- 
Rotative Pressure Engines. 

An effective method of obtaining an advantageous action of 
the steam is to substitute for the reciprocating mass of the 
Cornish engine a rotating mass. This is accomplished by 
using the reciprocating motion of the piston to operate a crank 
shaft upon which a fly-wheel is placed. Since it is practicable 
to give the rim of the fly-wheel four to six times the velocit}' of 
the crank pin, the magnitude of the moving mass can be much 
smaller, and since the value varies as the square of the mean 
velocity, the mass is reduced at least 16 times. It is therefore 
possible by this means to give even small pumps an efficiency 
equal to that of large pumping engines. J 

It is not practicable to construct single-acting pumping 
engines into fly-wheels, because the piston speed v is too varia- 
ble. If we draw a curve representing v, the ordinates being 
the positions of the piston, we have for a connecting rod of 
infinite length a circular curve, as in Fig. 102 1 a. When the 



Fig. 1018. 



Fig. 1018 shows three devices for this purpose. At a is shown 
a closed system with pistons of equal diameter ; i is a similar 
one with unequal pistons ; and c is a form with combined 
pistons. Such water-rod connections are adapted for use in 
mines, and the following example will illustrate. 



/ 


T^ 


i ,><f^^^T^ 


^«5> 


V 1 .-es'^^ 


? 


I \ 


^)1/^ : Vi 


XU- :' 


/ 


f 


i f\ 

i fmN 

! : \ 

t : 


fr^\ 


I 




il 2 


:i 





.Jx^ 








Lu'S 


f^-ft^ 


L.:-. 


L.* 


^■--^-■'■ 


V 




\ 


f 


\ i / \ 

'■-V 






3" 


V- i 




Fig. 1019. 



The arrangement of transmission in the Sulzbach-Altenwald 
is shown in Fig. 1019, which represents the engine above ground, 
while Fig. 1020 shows the mechanism in the mine. 




Fig. 1021. 

length of the rod is taken into account these curves are modi- 
fied, as shown in Fig. \02\b, which is drawn for a rod four 
times the length of the crank. This curve also shows the ratio 
of the tangential force on the crank pin to the pressure on the 
piston. \ 

The variations in the value of v, which often differ widelj- 
from the mean value Vm^ must necessarily be communicated to 
the mass of water, and hence great variations occur in the 
stresses. For this reason the velocity of the column of water 
must be kept within moderate limits, notwithstanding the use 
of air vessels. These variations become much less serious 
when two pumps are connected b}' cranks set at right angles 
with each other. The corresponding velocity curve is shown in 
Fig. 1021 c, and many pumping engines are now so made. More 
recently triple cylinder engines are made with cranks 120° 
apart. The velocity curves in this case are shown at d. It is 
evident that both these forms involve complications in con- 
struction which compare unfavorably with the direct-acting 
pump with equalizing cj-linders (see ? .125). 

Instead of uring a revolving fly-wheel, the mass of metal may 
be arranged to swing in an arc of a circle of large radius. An 
ingenious application of this principle has been made by Kley, 
in his water works engine with auxiliary crank motion. The 
proportion between the steam pressure and the vibrating mass 
is so arranged that the auxiliary crank comes to rest either a 
little before, or a little be3-ond the dead point, so that the re- 
turn stroke in each case can be effected by the action of a 
cataract. In the first case, the flv-wheel swings backward after 



Fig. 1020. 

The arrangement is of the same form as Fig. loiSi. The 
steam piston c operates the two plungers i, b.,, which in turn 
operate the plungers Cj c/, and c, c/ in the mine, the pump 
plungers e^ e.^ being placed in the middle f 



*See Engineering, Vol. XLI., 18S6, p. 47, also H D. Pearsall, Principle of 
the hydraulic ram applied to large machinery. London. 18S6. 

t See Zeitschriftfiir Berg, Hiitten und Salinenwesen, XXII. p. 179 ; XXIII., 
p. 6; XXIV., p. 35. The depth is 820 feet, the speed from 6 to 16 double 
strokes per minute, with a pause of one second, giving about 420 feet piston 
speed per minute. This engine, built by the Bayenthal Machine Works at 
Cologne in 1858. has operated regularly for 29 years without any interrup- 
tiou worth mentioning. 



X The Gaskill pumping engine is a duplex pump with fly-wheel, and 
cranks at right angles, and has given excellent results. See Porter's " Re- 
port of the Gaskill Pumping Engine at Saratoga." 

V 

g Referring to the designations ni Fig. 1022, we have — = sin u) -t- tan a. 

cos t,). Since Pdz ^ 
P' rdui and Pv= P' c, 

P' 
the ratio —7^ is also 

equal to the same ex- 
pression. Hence the 
curves above given 
also show the ratio of 
the force in the path 
of the crank-pill to 
the pressure on the 
piston. 




Fig. 1022. 



In Fig. 1022 ti and S, P and c are represented by 1.2'; in Fig. c, by 2' 2"; 
in Fig. 'i, by 2' . i' : in c and d, the ratio of connecting rod to crank is again 
takeii as infinitely great. The curves are adapted lor double-acting pumps. 
When two single-acting pumps connected to right-angled cranks are used, 
the second half of the cun-es of Fig. i become the same form as the first. 



234 



THE CONSTRUCTOR. 



the pause, and in the second case, forward." The valve motion 
of this form of engine is considered in the following section, 

? 328. 
Valve Gears for Rotating Engines. 

Rotative engines are distinguished from pure reciprocating 
pressure organ escapements in that the}' deliver their effort in 
the form of rotary motion adapted to be used for driving run- 
ning machinery. Between the two forms there is also the 
intermediate kind, with merely auxiliary rotative mechanism, 
such as have been already referred to. The translation of 
reciprocating and rotary motion may be accomplished in a 
variety of ways, but by far the most useful and best known is 
that by which the rectilinear motion of a piston is transmitted 
to the shaft by crank connection. 

The variations in the tangential component of the pressure 
P' on the crank pin, Fig. 1021, becomes still greater when the 
pressure P, on the piston also varies by reason of the expan- 
sion of the steam. For this reason some form of equalizer is 
required in the form of a fly wheel. This latter becomes a 
reservoir for the storage of living force. Extreme examples of 
this action are found in rolling mill work in which within a 
brief time a looo H. P. engine may be called upon to deliver 
2000 H. P., a demonstration of action of the flywheel as a 
reservoir of power. 

The valve gearing for rotative engines is an important ami 
extensive subject. In the preceding sections a series of valve 
gears have already been described. These have all been based 
upon the principle of operating the valves by a direct recipro- 
cating motion, taken either from the piston or piston rod. 
With rotative engines another method is used, the motion beini; 
taken from the revolving portion of the machine, and this 
method may also be adopted for pumps with auxiliary crank 
action. We may then distinguish between : 

Reciprocating valve gears, and 
Rotative valve gears. 

Rotative valve gears are desirable even for direct acting 
pumps, but in a still greater degree are they desirable for rota- 
tive engines. Watt's rotative engine was made with a recipro- 
cating valve gear.t and this form has one advantage in that it 
is adapted for rotation in either direction. 

Hornblower, the inventor of the compound engine, also used 
a reciprocating valve gear. The slide valve, .invented by Mur- 
dock, in 1799, led the way to the introduction of the rotative 
valve gear in iSoo, but the old reciprocating gear still continued 
to be used, and is even re-invented at the present time. The 
later direct acting steam pumps with auxiliary rotative mechan- 
ism are almost always made with rotative valve gear. Kle}''s 
pumping engine, referred to in the preceding section, is made 
with reciprocating valve gear, since its motion is both before 
and behind the dead points of the crank. 

The use of the slide valve, combining four valves in one mem- 
ber, enables a very simple valve gear to be made for the ordinary 
double acting escapement, as the diagram of a plain slide valve 
engine, Fig. 1023, clearly shows. 





Fig. 1023. 

The use of an eccentric r, and rod /, to operate the valve (5. 
is not the earliest form of gear, the first method being the use 
of an irregularly shaped cam which brought the valve to rest 
except at the time of opening or closing.^ A feature of the 
slide valve which was long overlooked was the fact that the 
time of closing the steam ports //and /// could be regulated 
so as to effect the proper expansion of the steam. In order to 
accomplish this result without impeding the exhaust of the 
steam, the eccentric r, must be given the so-called angle of 
advance 2° i . 2' beyond the mid-position. The direction of rota- 

*For a fuller account of this interestiiio; engine (German Patent No. 2345), 
of which more than fifty are in operation, see : Eerg-u. Hiittenm. Zeituny 
Gliickauf, 1S77, No. 18, 1S7Q, No. gS ; Mor.iteur des int. matdriels, 1877, No. 20°; 
Compt. rend, de St. Etienne, 1S77, June ; Bergg-ei.st. 1S79, No. 85 ; Z. D. Ingen- 
iure, 1879, p. 304, iSSi, p. 479, 18S3, p. 579. Duigler'.<i Journal iSSi, p. i, i8S=, 
p. 349: Maschinenbauer 18S1, p-63 ; Oesterr. Ztg. f. Bersr u. Huttenwesen 18S2 ■ 
Kohleninteres.sent (Teplitz) 1S82, No. 34 ; Revista metalurgica (Madrid) 1SS3, 
No. 968. 

tSee Farey, Treatise on the Steam Engine, London, 1S27, p. 524. Engines 
with slide valves were only made by Boulton and Watt, aft»r James Walt 
retired to private life. 

J See Farey, p, 677. 



tion of the crank is then governed by this angle, the arrange- 
ment above giving rotation to the left, and the position i 2"" 
for ^i, giving right-hand rotation. 

The action of the slide valve may readily be represented 
graphically.? The angle of advance and lap being given the 



point of cut-off can be determined by the following method. 




Fig. 1024. 

Fig. 1024. The circle i Q represents the circle of the eccen- 
tric and may also be taken as the crank circle on a reduced 
scale. C" and C" are two symmetrically placed positions of 
the piston at which it is desired that the cut-off shall take place. 
Through these points with a radius 1.3 = / describe arcs from 
centres 3" and 3"' ; their intersections E^ and E^ with the cir- 
cle give the angles at which the expansion Q C" and C C" 
occurs, in this instance j'j of the stroke. We now select the 
point I'o of the crank circle at which the admission shall begin, 
join V^ E^ and draw the equator 2.1.2' parallel to it, and the 
angle 2 . \ . C will be the angle of advauce S, and the distance 
of 2 . I from E2 l-'-i, tlie outside lap t'.j for the port //. The 
width of port a must also be chosen, and must be so taken that 
it is less than r, — ■ e.^, and is represented by the parallel ^.,. 
When the crank reaches /,, in this instance at ^^'f, of the stroke, 
the exhaust begins, and the distance i^'i, of the, parallel I., I^ 
from the equator is the inside lap. 

The construction is similar for the other half of the stroke. 
The angle S is already known, and hence the parallel /i3 V^ 
from E^ can be at once drawn, and the admission point K, de- 
termined. The outside lap c^ is somewhat less than e,,_, thus 
giving a correspondingly wider port opening. The inside lap 
73 is made equal to i.,, and the bridges b^ and b., are made equal, 
thus giving a symmetrical valve seat. A certain amount of dis- 
cretion is permissible in the selection of b^_^= b-^; care being 
taken that there is sufficient bearing at the extreme valve stroke 
to insure tightness. The points // and // are also of impor- 
tance, as the}' determine the closing of the exhaust. The cor- 
responding piston positions Of and Cfare not symmetrical,, 
because i-^ = 7',, but the inequality in the compression is not 
serious. 

The above method of considering the influence of the ratio 



— is very simple. It is easy to substitute any desired ratio — ' , 

but the variation is slight. It must be noted that the distance 
I . 3 must be laid out to the actual scale of construction. 

I The application of 

Ed i Zeuner's diagram to 

the same case is made 
in the following man- 
ner, Fig. 1025. The 
circle i C^ represents 
as before, the eccentric 
circle and the crank 
piu path. The angle 
Co . I . 2 = C' . I . 2 = 
90 — (5. With I as r. 
centre, describe circles 
with radii (?and 7', here 
made alike for both 
ends of the valve, also 
one of radius e + a. 
Upon I . 2 and l . 2 as 




Fig. 1025. 



diameters, describe circles, called the valve circles. 



g Formerly the so-called " valve ellipses " were used ; since i860 Zeuner's 
diagram has superseded these, see Zeuner, Schiebersteuerungen, Freiburg, 
Euglehardt, first published in Civil Ingenieure, Vol. 2, 1856. 



THE CONSTRUCTOR. 



23S 



The intersection of radii from i, with these circles, give the 
distance of the valve from its middle position for various crank 
positions. For the position i f'j, for instance, the admission 
for the left stroke begins, at i E^_ the expansion, at i / the ex- 
haust, etc.* 

The Zeuner diagram gives the valve position by means of 
polar co-ordinates, while the writer's diagram is based on par- 
allel co-ordinates. To be strictly correct, the valve circles i . 2 
and I . 2' of the Zeuner diagram should fall upon each other. 
The arrangement shown has been adopted by Zeuner as more 
convenient in practice. 

It will be seen from the preceding that the, rate of expansion 
can be varied by altering the eccentricity and the angle of ad- 
vance. This may be carried so far that the direction of rotation 
is changed, giving what is termed a reversing motion. A variety 
of reversing motions have been devised, which accomplish the 
desired relation of parts by shifting a reversing lever. Ot these 
the most practical are the so-called link motions, of which a 
number will here be briefly shown.f 




Fig. 1026. 

Fig. 1026 a, is an outline diagram of Stephenson's link 
motion. The link 3' i" , of convex curvature towards the valve, 
is given an oscillating motion by means of the two equal eccen 
tries I . 2', I . 2", and is suspended from its middle point 7, 
from the bell crank lever S 7'. The motion of the link is trans- 
mitted to the valve by means of the sliding block 5, and rod 6. 
Fig. 1026 b, is Gooch's link motion. The link 4 is driven by 
two eccentrics as before, but is curved in the opposite direction 
with a radius 5 . 6, and is suspended from its middle point 8 to 
a fixed pivot 8', while the rod 5 . 6 is shifted by means of the 
lever connection S 10 . 10'. 




Fig. 1027 <J, is the link motion of Pius Fink. In this form 
the link is operated by a single eccentric instead of two, as in 
the previous forms. This simple mechanism is not as widely 
used as its merits deserve. 

Fig. 1027 b, is the link motion of Allan, or Trick. In this 
design the link 4, is straight, and both the link and th. radius 
rod are suspended and shifted by the lever connections 8' . 8, 
and 9' . g.f 




Fig. 1028. 

Fig. 1028 a, is Heusinger's link motion. The link 4, vibrates 
upon a fixed centre 9, and is operated by an eccentric i . 2. The 
valve rod is moved from the main cross head by theconnectioiis 
10 . II . 6 . 7, and also by the radius rod 5 . 6, which latter is 
suspended from the bell crank S . i^' ■ 

Fig. 102S 0, is Klug's valve gear, known in England as Mar- 
shall's. The curved link 4, is rigidly secured and does not 



* It is usual to make the valve symmetrical, /. e., e^ = c^, which necessarily 
causes the cut-off to take place at different points for the back and forward 
strokes. 

I See also Zeuner, as above; Gustav Schniidt, Die Kulissensteurungen, 
Zeitschr. d. osten. Ing^. u. Arch.-Vereins, 1S66, Heft. II. ; also Fliegner, Ueber 
cine gelr. Lokomotiv-steuerungen, Schweiz. Bauzeitun^, March, 18S3, p. 75. 

X Sec Reuleaits, Die AUanische Kulissen steuerung, Civ. lug., 1857, p. 92. 



move. The eccentric i . 2, moves the valve connection 6. 7, by 
means of the lever 2.3.6, which vibrates about the point 3, on 
the end of the radius rod, the other end of the rod being held 
by the link block 5. Instead of the link 4, a radius arm 4^ 5, is 
often used, the centre 4^ corresponding to the centre of curva- 
ture of the link, the action being the same in both cases.? 





Fig. 1029. 

Fig. 1029 (2, is Brown's valve gear, which differs from the pre- 
ceding by the substitution of a straight link of adjustable angle, 
for the curved guide link. 

Fig. 1029 i, is Angstrom's valve gear. The point 3 of the pre- 
ceding gear is guided by a parallel motion, and the point 6 is 
between 2 and 3, instead of beyond. 

The eight preceding valve gears operate the valve approxi- 
mately in the same manner as if a single eccentric of variable 
eccentricity and angular advance were used, the eccentric rod 
being assumed of infinite length as compared with r. The path 
of the successive positions of the middle po'nt of this imaginary 
eceutric is called the central curve of the valve gear. 









Fig. 1030. 



Fig. 1030 shows the form of this curve for link motions in 
general. Form «, is that for cases i, 4 and 5 ; form b, for case 
I, when the eccentric rods are crossed, and form c, in which 
the curve becomes a straight Ime, is for cases 2, 3, and 6 to 
8. In the latter instance, the lead, or opening for admission of 
steam at the beginning of the stroke is constant, a point con- 
sidered by many to be of much importance. 

It is possible to arrange the mechanism in such a manner that 
the centre of the valve motion may move directly in the desired 
central curve, as is shown in Fig. 1031. 




Fig. 1 031. 



This construction involves the rotation of the link about the 
crank axis. The only point to be accomplished is to guide the 
centre 2' in the path 2' . 2 . 2." . Fig. 103 1 r, is a direct guide 
for the eccentric with wedge adjustments ; h, is Sweet's valve 
gear, in which the position of the eccentric is determined by a 
centrifugal governor.|| This only uses the central curve from 



g For a further account of this gear, see : Berliner Verhandl, 1S77. p. 345, 
1882, p. 52. Engineering, Aug. 13, Oct. i, Dec. 3, 1880 ; Nov. 4, 1881 ; June 23, 
1882; Feb. 6 and 27, 1885; Jan. 12, i836 ; Sept. 9, 1887. Engineer, May 26, 1S87 ; 
Feb, 23, Mar. 30, April 27, June 29, 18S3; June 5, 1S85, Marine Engineer, 
1885, No. I,, Civ, Ing, Heft, 7 and 8, 1882 ; Zeitschr, D, Ing, 1885, p. 280, 1886, 
pp 509-625 ; Revue universelle, 1S82, p 421; Busley, Schiffsraaschine, I , p. 
454; Hart^mann, Schiffsmaschinendienst, Hamburg, 1884, p, 53; Blaha, 
Steuerungen der Dampfmasch, Berlin. 1885. p, (i^. 

II See Rose, Mech, Drawing Self-taught, Philad, Baird ; for similar gears. 
see Am, Machinist, Grist, Oct. 5, 18S3 ; Ball, ditto -Aug, 18, 1883 ; Harmon, 
Gibbs & Co., ditto Nov. 24, 1883, .'^Iso Sturtevant, The Engineer, New York, 
Jan. 1888. 



236 



THE CONSTRUCTOR. 



2' to 20, and tiie path is a curve produced by a radial arm as in 
Klug's valve gear. The valve is balanced, in order to reduce 
friction to a minimum. 

The last described valve gear possesses the advantage of great 
simplicity but retains the disadvantage of all single valve gears 
wheu used for a high expansion ratio, /. e., the admission and 
exhaust of the steam do not remain uniform, and are often un- 
satisfactory. For this reason manj' valve gears with indepen- 
dent expansion valves have been designed. 




^^- 



Fig. 1032, 






Three forms of gear with separate expansion valves are shown 
in Fig. 1032. The form a, is known as Gonzenbach's ; that of 6, 
by various names; c, is the widely used Meyer valve gear.* 

In France, Farcot's gear is used, having two loose cut-off 
plates carried on the back of the main valve, and in America, 
the excellent Porter-Allen engine with double valves operated 
by two eccentrics, is much used. Rider's valve gear is a modi- 
fication of Meyer's, Fig. 1032 r. The two cut-off plates form re- 
verse spirals, aud slide in a concave seat on the back of the 
main valve, the admission parts being also spiral shaped and 
cut-off varied by twisting the cut-off valve axiaiiy-t 

Instead of using eccentrics to operate the valves, cams of 
irregular outline may be adopted, these permitting a rapid 
opening and closing of the parts. Noteworthy examples of 
cam valve movements are to be found in the steamboats of the 
■western and southern States in America. In its original form 
of a cock or cone a slide valve may be operated by an alternat- 
ing motion as well as by continuous rotation. Such valves 
have been used in steam engines by various builders, among 
them the firm of Dingier in Zweibrucken, but the cost prevents 
wide use. A most extensive use of oscillating cylindrical 
valves has been developed by Corliss and his followers. 

The forms of oscillating aud rotating chamber gear trains 
already described involve other means of operating the valve 

than are used for recip- 
rocating engines, and 
shown in Fig. 1023. As 
an example, the water 
pressure eugine of 
Schmid, of Zurich, Fig. 
1033 is given. In this 
instance the valve b, is 
formed in the frame of 
the machine, and is of 
the type shown in Fig. 
987 c. The regulation 
of speed of rotative 
water pressure engines 
is a much more difficult 
matter than is the case 
with steam engines, 
partly on account of 
the lesser fluidity of 
the water and also be- 
cause of its slight elas- 
ticity. An air chamber in the admission pipe as shown in Fig. 
1033 is therefore desirable, and when extreme changes of load 
are anticipated the valve gear should be modified. If it is 
desired to cut off the admission of water before the end of the 
stroke is reached, it is necessary to arrange a special valve to 
permit the discharge to continue. Excellent engines with this 
arrangement have been made by Hoppe, of Berlin, for the 
Mansfeld mines, and for the Frankfurt railway station. 

Another method is applicable to power driven pumps, an 
illustration of which may be found in the design of Franz Hel- 
fenberger, of Rorschach.! This is made with a hydraulic 
ratchet mechanism arranged in the crank disk in such a man- 
ner as to move the crank pin to or from the centre, the ratchet 
being operated by tappets which strike each time the crank 
passes the dead centres. The throw of the crank is thus varied 
to correct for variations of speed, the mechanism being con- 
trolled by a regulator. The action is very satisfactor}', giving 
results varying from 90 to 82 per cent, for a change of power 




Fig. 1033. 



from I to I, according to the investigations of Autenheimer, 
Buss and Kuratli, in 1885. J 

A later device is that of Rigg, which also acts by varjdng the 
stroke. The machine is a so-called "chamber crank train" 
(described in Theoretical Kinematics, p. 359: English ed., p. 
361), with four single acting cylinders carried on the revolving 
wheel in the same manner as the machines of Ward, Schneider 
aud Moline. The length of stroke is controlled by a regulator, 
similar to Sweet's governor. Fig. 1031 b, which operates a hy- 
draulic escapement and adjusts the radius. This device is used 
by Rigg for steam and air eugines to control the degree of ex- 
pansion. These latter machines are operated as high as 2000 
revolutions without producing trembling. j| 

Besides the various forms of valve gear which have already 
been described, there are also the numerous "trip" gears, of 
which some examples have been given in ? 252. These gears 
are made in many forms. The valve is made in four parts, as 
indicated in Fig. 9S6, on account of the facility with which the 
release can be controlled by the regulator. 

The varieties of trip valve gears are most numerous, and 
there can be little doubt that the subject has been overdone, 
when it is considered that in many instances the entire mechan- 
ism of the eugine has no other aim than to determine the open- 
ing and closing of the valves. In America, where this form 
originated, the reaction has already set in, and there is a dis- 
position to return to the single slide valve, especial care being 
taken, however, to secure relief from pressure and to produce 
correct motion. 

There is to be found in some parts of Germany a form of 
valve gear which may be called an "inner ' ' and " outer " gear. 
This form does not pos- 
sess sufficient merit to 
meet general application, 
but may be briefly no- 
ticed. The mechanical 
action of parts is not 
different, whether the 
"inner" or ''outer'' con- 
struction is used, and 
either arrangement may 
be adojited, at the dis- 
cretion of the designer. 

The following exam- 
ples will make the ar- 
rangement apparent, as 
well as the illustration 
already given of Schmid's 
water pressure engine, 
Fig. 1033. Fig. 1033 a is 
an "outer" valve gear 
for a blowing cylinder, 
and Fig. 1034 6 is a valve 
for a vacuum pump.TJ 
Another example is Cu- 
velier's valve, which is 
placed entirely outside of 
the steam cylinder, as is 
also the gear of Leclerq.'^* 
The ordinary slide valve 
is partly' without and 
partly within the ma- 
chine, being outside the cylinder and within the steam chest. 




i°34- 



C. ADJUSTABLE POWER ESCAPEMENTS 
§329- 

Adjustabi,e Pump Gears. 

The principles of adjustable escapements have already been 
discussed in ^ 259, and examples of rigid construction given. 



*An excellent gear with two valves operated by a single eccentric, is 
Bilgram's. See Bilgrara, Side Valve Gears. Philadelphia, 1S78. 

t This is an excellent problem in kinematics, the action of the valves 
and spiral ports forming a kinematic chain. See Theor. Kinematics, p. 333. 

J German Patent, No. i2,oiS, Jan. 27, 1881. 



g A third sj'stem is that devised by Hastie, of London (see Engineer, 
August. 1878, and April, 1880, p. 304). Hastie controls the position of the 
crank pin by means of a pair of curved cams which act against increasing 
external resistance, aud opposed by a spiral spring ior diminishing resis- 
tance. This device does not give complete regulation for the following rea- 
sons: I. In order that the statical moments of the increasing resistance 
and driving force on the crank pin shall be equal, the spring must act on 
the curved cams in such a manner as to cause the crank pin to move out- 
ward. This can be approximately accomplished by a careful arrangement 
of parts, but only approximately. Under this arrangement, however, there 
is a tendencv for the pin also to move outward when the driving force in- 
creases, instead of moving promptly inward as should be the case. An 
attempt to correct for this error, reverses it. 2. The angular velocity of a 
motor is not a function of the impelling force, ?'. e., the machine will run 
fast or slow as the case may be, this fact also appearing in practice. This 
is a common error into which inventors of " dynamoinetric governors" 
have fallen, even Poncelet, himself, havijig done so in his dynamometric 
regulator. 

II See Rigg, Obscure Influences of Reciprocation in High Speed Engines, 
Trans. Soc C. E., 1886 ; also Engineer, June 4, 18S6. These engines are used 
oti the compressed air .systems of Birmingham and Leeds. 

If See Oppermann, Portfeuille econ., Feb. 1S8S, p. iS. 

**See Genie ind. 1864, also Schweiz. polyt. Zeitung, 1864, p. 83. 



THE CONSTRUCTOR. 



237 



Their action consists of the two following operations : i. By 
the adjustment of one part the release of another part to the 
action of an impelling force is accomplished ; 2. By the attain- 
ing of motion of the checked member, the checking is again 
produced either directly or indirectly. 

These principles also obtain for escapements for pressure 
organs, and inchide a great number of important applications. 



pressure being supplied from an accumulator. The valve b is 
operated by the plunger b' against the pressure of the springs 
b"' , and again reversed by the pistons Cj C and connection 5. 
The piece at 6 is not a lever but is a cross head connected with 




Fig. 1035. 

The general scheme of such an adjustable gear for a steam 
pump cj'linder is shown in Fig. 1035 a. The valve chest d^ is 
made separate from the cylinder d, and is capable of movement 
parallel to it, the connections a^ a„ being made flexible. The 
slide valve b is operated from above by the lever b' . When the 
lever b' is lifted the pressure is admitted under the piston c 
through the port ///, while the space above the piston is in 
communication with the exhaust //'. This causes the piston to 
move upwards and hence the lever c' moves the steam chest d-^ 
also upwards, thus closing the valve ports and checking the 
movement of the piston. If the lever b' is again lifted this 
action will again take place, and so on until the upper limit is 
reached. A reverse motion of the valve lever produces a cor- 
responding reverse motion of the piston.* The same action 
may be obtained by using the arrangement shown in Fig. 1035 b. 
In this case the valve chest is fast to the steam cylinder, while 
the valve is so arranged as to be moved both by the hand lever 
b' and by the piston rod c. When the valve is moved by b' , the 
piston also moves and closes the valve by the lever c' , thus 
bringing itself to rest again. The piston r follows the motion 
of the lever b' in either direction ; starting when the lever is 
started, and stopping when it is stopped. Any resistance not 
exceeding the force of the pressure at a,, can thus be overcome 
while the resistance to the operator is only that due to the frac- 
ture of the valve and connections. The practical value of this 
device in many directions will be evident, and the examination 
of the above simple forms will explain the action of the various 
modifications. 

Two constructions, designed by the author in 1866 for regula- 
tors will be found described in the Civil Ingenieiir. f The 
lever is connected with the valve by means of a double parallel 
motion which is moved by the piston motion back into a posi- 
tion parallel with the base line. 

The operation was satisfactory but the apparatus was cum- 
brous. In iS58 Farcot constructed a similar device, using an 
approximate parallel motion, but the apparatus was too com- 
plicated to be practically satisfactory. j A somewhat simpler 
construction was afterwards made by Farcot, but this was also 
too complicated for practical use.? Other designs have been 
made by Farcot. Some recent constructions are here given. 

Fig. 1036 shows the hydraulic steering gear of Bernier-Fon- 
taine & Widmann,|| which is similar in principle to Fig. 1035 b. 

In this case the controlling gear b' , consists of a small hy- 
draulic piston. The water pressure is admitted to it through 
the pipe a' , and is opposed by the spring a" . The two plun- 
gers Q and C,, act as a double acting piston, the hydraulic 



* The escapements shown in \ 259 are only single-acting, and do not admit 
of a reverse motion. 

t Civ Ingenienr, 1879-1880, Prof. Rittershaus, Ueber Kraflvermittler. Also 
the models at the Roval Technical High School in Berlin. 

tSee Civ. Ingenieur, 1S79, i83o, Rittershaus, Ueber Kraftvermittler. 
(Intermediate Power Mechanism.) A model of the design of 1S66 is in the 
cabinet of the Royal Technical High School at Berlin. 

§See Aiinales industtielles 1873, p. 518 ; also Oppermann, Portfeuille econ. 
1874. p. iij- 

J See Revue Industrielle, if 86, p. 373; 18S7, p. 148. 




Fig. 1036, 

the valve. The admission and release of water pressure through 
a' forms a long distance transmission involving the use of an- 
other escapement ; the whole thus forming a.gear of the second 
order. 

Fig. 1037, is a neat regulator for steam engines by Guhrauer 
& Wagner. 1[ In this, as in 
Fig. 1035 a, the valve seat is 
capfible of movement iu a 
direction parallel to the pis- 
ton f, and is made concen- 
tric with the piston rod, the 
valve b, being a piston valve 
or rod moved by the gover- 
nor. The piston c is subjec- 
ted to full steam pressure 
from (?i on both sides through 
the ports II and III, but as 
soon as the valve b is moved 
up or down, the holes b^ re- 
lieve the pressure on one 
side or the other, the equi- 
librium is disturbed and the 
wiredrawing of the steam 
through the small ports II 
or III prevents sudden action 
and the piston moves until 
the holes are closed. As 
might be expected, this de- 
vice is very satisfactory in 
practice. 

Devices of this type are 
well adapted for steering 
gears as well as for regula- 
' tors and a very delicate ap- 
plication of the principle is 
found in the Whitehead tor- 
pedo, in which the steering 
gear which determines the 




Fig. 1037, 



depth of the torpedo beneath the water is thus controlled by a 
barometric device. 

I- 330- 

Adjustable Gears for Rotative Motors. 

The principles of the gears described in the preceding section, 
are also applicable to rotative motors although the arrangement 
is not so simple as with direct reciprocating cylinders, since 
the motion of the valve gear has also to be controlled. At the 
same time it must be noted that the application of adjustable 
gears to direct acting reciprocating motors is the more recent 
of the two. The earliest rotative gear of this sort, so far as the 
author has been able to ascertain, is that designed by F. E. 
Sickles, of Providence, R. I, in i860 (See also ?. 252).** 



1[ Built by Ganz & Co , of Budapesth, with Meyer's and with Rider's valve 
gears. 

** According to the catalogue of the Centennial Exposition, in Philadel- 
phia, in 1876, Vol. II. p. 52, Sickles made his first application in 1849 , his 
patent was granted in i860, and his first machine exliibited at the London 
E.^hibition of 1862, 



238 



THE CONSTRUCTOR. 



Sickles' machine was made with two oscillating Cylinders. 
Both eccentrics were fastened together and were loose on the 
crank shaft and operated by a hand wheel and spindle. The 
steam chests oscillate with the cylinders. The crank shaft re- 
volved in the same direction as the hand wheel is turned, but as 
soon as the motion of the latter was stopped, the valve seat 
moved under the valve aud the ports were closed. 

The more recent forms of adjustable valve gears for rotative 
engines are made after two distinct and important principles. 
The first form is that in which a double engine, without a fly 
wheel aud with ordinary slide valve gear without angular 
advance is used, in order to permit rotation in either direction. 
The ports I aud IV are then made so as to be interchangeable 
so that I can be connected either with the admission a^ or ex- 
haust a„ ; and IV with the e.^haust a., or admission a^, at will. 
This change of connection is effected by means of an auxiliary 
valve sometimes known as a "hunting valve." This hunting 



ilib; 




Fig. 1038. 

valve can readily be controlled by hand for a steering en- 
gine, for which it is well adapted, since the angular motion of 
the rudder pin is limited, seldom exceeding 90°. The adjust- 
ing valve can then be arranged according to either of the prin- 
ciples of Fig. 1035 a or b. The following designs will illustrate 
the construction. 

Fig. 1038 shows the steering gear of Dunning &. Bossiere.* 
The adjusting valve b rides upon a moveable valve seat bo. The 
lower port A is always in communication with chests of the 
two engine cylinders, while the upper port J is in communica- 
tion with the central port under the valves. The lever b' is 
connected with the spindle b" by an internal gear. This spiudle 




Fig. 1039. 



Fig. 1040. 



has a screw thread of steep pitch, and is connected to the ad- 
justing valve b. The moveable valve seat bo is connected to a 
spindle bo", which has on it a much slower screw thread, and 
is also geared by bevel wheels to the axis c' of the drum of the 
tiller chains. Whenever the engines are started by moving the 



lever b' , the chain drum revolves and shifts the moveable seal 
bo until the ports are again closed. The parts are so propor- 
tioned that the angle through which the rudder is moved is 
equal to the angle through which the lever b' has also been 
moved. This enables the position of the rudder to be deter- 
mined at once by noticing the position of the adjusting lever. 
The moveable valve seat bo will be recognized as the same in 
principle as the moveable steam chest of Fig. 1 035*7. The 
spindle b^' can be prolonged to operate an indicator on the 
bridge for the inspection of the officer in charge of the ship.f 







Fig. 104 1. 

Fig. 1039 shows Britton's steeling gear.j The adjustment is 
effected by a hand wheel and screw b' operating the lever b" at 
6, and thus moving the valve b. At 7 this same lever is con- 
nected to a nut on a screw thread cut on the axis c' of the chain 
drum, so that the motion of the latter closes the valve after it 
has been opened by b' . This corresponds in principle to Fig. 
1035 b. 

Fig. 1040 shows the steering 1 

gear of Douglas & Coulson.J 
This is another application of the 
same principle as the preceding 
device. When the adjusting screw 
b' moves the nut, lever and rod b 
out of the mid-position, the re- 
volving axis c of the chain drum 
turns the nut A/ by means of the 
spur gearing until the dead posi- 
tion is again reached. 

Fig. 1041 is a steering gear de- 
signed by Davis & Co.|| This is a 
simple application of the principle 
of tig. 10356. The hand whet 1 
shaft b' has a screw thread at 6, 
the nut being in the hub of the 
worm wheel c' , the latter being 
driven by a worm on the crank 
shaft. Any adjustment of the 
valve rod b by turning the hand 
wheel results in a corresponding 
readjustment by the motion cf 
the worm wheel and nut derived 
from the engine. 

The second kind of adjustable 
gear for rotative engines is much 
less frequently used than the first 
form. In this arrangement the 
adjustable valve is not connected 
with the main valve gear, but is 
operated independentl3', so that 
the crank will make any desired 
number of turns in either direc- 
tion, according to the motion 
which is given to the adjusting 
valve. 

Fig. 1042 shows Hastie's steer- 
ing gear.l[ This is based on the 
principle of Fig. 1035(7. The mov- 
able valve seat b,^ is operated by 
the piston c, the eccentric c^ being 
so placed that b^ has a reduced motion coincident with that of 
the piston c. The valve b is operated by the eccentric b", which 




Fig. 1042. 



• See Revue Industrielle, 1866. p. 401. 



t A steering: gear of sirailar design, with moveable valve seat bo and ad- 
justing valve, is that of Hastie, English Patent No. 1742, 1875. Also that of 
Holt; see Engineer. Sept , 1877, p. 221. 

X See Revue Industrielle, 1SS4, p. 435. 

S See Engineering. April, 1882, p. 281. 

ll See Engineering, April, 1882, p. 398. 

1[ See Rit^ershaus in Civil Ingenieur, already cited. 



THE CONSTRUCTOR. 



239 



has the same throw as c-^, and is moved by a hand wheel ou b' . 
The action which follows is that the crank shaft follows the move- 
ment which is given to the hand wheel both in direction and 
in revolutions. This action is similar to that of duplex pumps. 
Any number of revolutions may be made in either directions, 
and the device is a genuine rotative gear, as was also the ear- 
liest type, 2. e. Sickles' gear, already described at the begin- 
ning of this section. It would not be difficult to design a simi- 
lar gear on the principle of Fig. 1035 b, but the author has no 
knowledge that this has been attempted. 

Adjustable valve gears for rotative engines have generally 
been designed for steam steering engines, and in some of the 
recent powerful marine engines they have also been used to 
shift the link motion. There are many other applications 
which might be made. The speed can also be controlled by the 
adjusting wheel or lever b\ if desirable, b}' suitable connections 
to the steam valve. As simplicity in construction is most im- 
portant, steam economj' is not considered in designing ma- 
chines of this kind. 

D.-ESCAPEMENTS FOR MEASUREMENT OF VOLUME. 
I 331- 

Running Measuring Devices. 

In the classification of running mechanisms operated by 
pressure organs, it was noted that these devices could be used 
for measurement of volume. As already shown in § 321, fluid 
escapements are better adapted for measurements of volume ^ 
than for measurement of time ; but there is a close resemblance ' 
between the two operations, and many fluid meters might 
properly be classed as time-pieces. When the fluid to be meas- 
ured is a homogeneous liquid, the quantity and volume are in 
direct proportion. With fluids which are not homogeneous, 
such as gases and the like, a knowledge of the density is neces- 
sary in order to determine the quantity from the measured 
volume ; if the density is also to be determined at the same 
time as the measurement, the problem becomes much more 
complicated, as will hereafter be seen. 

Liquids are frequently measured by means of continuous 
runnnig devices; but the choice of construction is very limited. 
Among the open wheel devices there is available practically only 
the form shown in Fig. 957 d, and that onl3' when the liquid is 
under moderate pressure. If, then, the liquid is slowly con- 
ve3'ed off below the horizontal plane of the axis so that the 
acceleration of the wheel is uniform, then will the continuous 
rotation of the wheel be proportioned to the volume of the 
liquid passing through it. 

An instance of this construction is the measuring drum in 
Siemen's meter for spirits.* This is made with three buckets, 
and has inward deliver}'. Since the question of the density is 
in this case important, Siemens has devised a very ingenious 
float arrangement by which the counting mechanism is regu- 
lated to the volume of flow. 

When liquids under high pressure are to be measured by 
such a device, the wheel must be inclosed in a case in which 
also a gaseous fluid must also be contained under a correspond- 
ingly high pressure, which is usually inconvenient. For meas- 
urements of high-pressure liquids, the chamber gear trains 
already described are preferable, especially since it is practica- 
ble to pack the working joints reasonably tight.f Wheels in 
which the living force of the water acts are also adapted for 
this service, either as bucket wheels or in the form of reaction 
wheels (see ? 315). Siemens has constructed a meter of this 
kind, in which a reaction wheel is used, and the error of which 
does not exceed 2 per cent., plus or minus, j 

Another form, giving fair results for open channels, is based 
on Woltmann's fan. 

Gaseous fluids of small and only slightly varying density 
can be well measured by modifications of bucket water wheels ; 
the conditions being practically reversed from those already 
considered, and the water now being the surrounding medium, 
and the gas the one to be measured. 

One of the best known and most widely used devices for this 
purpose is the ''wet" gas meter of Clegg and Crosley, shown 
in Fig. 1043 a. The revolving drum is a wheel with four 
buckets, which is driven by the passage through it of the gas. 
The gas is introduced just above the horizontal plane through 



the axis, and the liquid used is water ; or, if there is danger of 
freezing, glycerine may be substituted. If the level of the 
water is lowered through evaporation or leakage, the volume 
of gas passing through the meter at each revolution will be 
increased, and to avoid this a float is so arranged that the gas 
v;ill be shut off if the water level falls too much. 



* See Zeitschr. D. Ing., 1874, p. 108 This meter i.s in extensive use in 
Russia, Sweden and Germany for measuring brandy and alcohol, and is 
very satisfactory. In Sweden, in 1^83, a comparison showed that a most 
careful hand measurement gave 15,365,931 litres of 50 per cent alcohol, and 
the meter gave 15,450,775 litres, or an e-xcess of only ^ of i per cent. 

t See the ^Jrown Water Meter in Schweizerische Bautzeitung, JIarch, 1883, 
p. 81 ; also Payton's Water Meter. 

X The older form {1S57, Z. D. Ing., p. 164) was on the principle of Segner's 
wheel. Fig. 962 a ; the niore recent design is made like the turbine of Fig 
963 d. At the end of 18S6 Siemens & Halske had made 88,500 meters, and 
the English house of Siemens Bros., 130,000 of the old and new patterns. 





FiG. 1043. 

For very accurate measurements of gas, Sanderson's meter is 
used. Fig. 1043 b in the water level remains unaltered so long 
as the vessel is kept supplied. The semi-cylindrical float is 
pivoted on the axis C, and is so constructed that the centre of 
gravity of all the sectors is the same as if the sheet metal body 
were homogeneous. If the float moves through an angle a 
with a sector A C B, the thrust of the sector A'C D of an angle 
180° — 2 a will pass through the axis, with force P' for the sector 
A'C B. The weights P of the two equal sectors A C B and 
A'C B act downward through their centres of gravity, and are 
also equal to P'. In order that there may be equilibrium, if 
for any chosen value of a, P' shall equal 2 P, the specific 
gravity of the float (assumed to be homogeneous) must be 
equal to one-half that of the liquid in the trough, z. e., = li for 
water, or = 0.63 for glycerine. 

The preceding meters have heretofore been used only for 
gases under low pressure, but are equally well adapted for 
gases at high pressures, such as compressed air for power 
transmission, simply by increasing the strength of the case. 
This has been done at the author's suggestion in connection 
with the compressed air system at Birmingham, as has also 
been the case with the "dry" meter described in the next 
section. 

Anemometers, used for measuring the flow of air, generally 
belong to that class of running devices which are driven by the 
living force of the pressure organ isee ? 315). They are usually 
screw turbines, or some modification of them. It is always 
necessary to take into account the stress of the gaseous me- 
dium, in order to obtain the desired measurement, since the 
apparatus only determines the volume.? 



Escapements for Measurement of Fi,uids. 

There exist certain defects in running devices when used as 
fluid meters, such as the journal friction, and in chamber ge.ar 
trains the surface friction, which render the results inaccurate 
for fluids of weak flow. For this reason piston meters have 
been devised, these also utilizing the power of the fluid, and 
for these the application of escapements is necessary. Meters 
constructed on this principle have been used especially for 
measuring water. Among these maj- be mentioned Kennedy's 
water meter, a form which has been extensively used.|| This 
is usually made with a vertical cylinder, the valve being a 
fourwav cock operated by a tumbling gear similar to that of 
Belidor's water-pressure engine (? 325). 

Jopling's water meter f is a piston escapement of the second 



§ Running devices may also be used to measure time as well as volume, 
and in fact the oldest constructions for this purpose, the clepsydra, the 
sandglass, etc., belong to this class. Escapement clocks were introduced 
only m the middle ages. There have been numerous recent attempts to 
make running time-pieces. (See Redtenbacher, Bewegungs-mechanismen, 
Heidelberg, 1861, p. 34, pi. 79; also Rijhlmann, AUgemeine Maschinenlehre 
I., Braunschweig, 1S62, p. 62 ) The problem is a difficult one, since it in- 
volves the construction of a running device which shall operate both with 
a uniform and a determinate velocity. Examples are found in the driving 
mechanism of astronomical instrnnients, in which the motion i.s transmit- 
ted by friction, controlled bv revolving fly-wheel devices. With these may 
be in'cluded the fan regulators for the striking mechanism of clocks, and 
similar applications. 

II See Revue Industrielle. i8S[, p. 205. 

f'See Z. D. Ing., 1.S57, p. 164; Maschinenbauer, Vol. XVI., 1881, p. 324; 
Technologiste, 1S82, Vol. 42, p. 95. 



240 



THE CONSTRUCTOR. 



order. There are two parallel horizontal double-acting cylin- 
ders, each operating the valve of the other. 

Fig. 1044 shows a section of Schmid's water meter. This is 
made with two single-acting pistons, each also being the valve 
of the other, and the whole forming with the crank connection 
an escapement of the third order. 




Fig. 1044. 

Escapement meters are also used for gaseous fluids. A very 
extensively used form is the so-called "dry" meter used for 
measuring illuminating gas. These have, in many cases, super- 
seded the "wet" meter, since the use of the liquid seal is 
avoided. In order to prevent friction, these meters are con- 
structed with flexible diaphragms instead of pistons, much like 
the diaphragm pumps shown in J 317. A good example is 
Glover's dry meter, which is an escapement of the second 
order connected to a crank shaft which operates the counting 
mechanism. The diaphragms are made of linen, made imper- 
vious to gas by a gelatine si?Jng. These meters do not show a 
higher degree of accuracy than the wet forms. 

I 333- 
Technolocicai< Applications of Pressure Organs. 

The applications of pressure organs for technological uses 
are not so numerous or important as those in which they act 
in connection with the help of various machines. These appli- 
cations are not dissimilar from those of tension organs, which 
have already been discussed in 'i 263. A general survey will 
be of value for the better understanding of the whole sub- 
ject. 

The use of a pressure organ from a technological standpoint 
consists in so using it that the result is a modification in form 
or shape either of another body directly by the action of the 
pressure organ or of the pressure organ itself by the other 
body. This "forming" action adds a fourth manner of action 
for pressure organs to those already classified in g 309, so that 
we now have : 

1. Guiding, 

2. Supporting, 

3. Driving, 

4. Forming, 

as the four methods of action or application. Of these the 
first three belong to all classes of machine elements used 
in construction ; the fourth falls within the domain of tech- 
nology. 

In order to speak comprehensively of the action of pressure 
organs, we will arrange them in five groups, according to the 
method of action, viz. ; by Filliu.u, Discharging, Internal Flow, 
Jet Action and Inclosing or Covering. 

a. Filling, 

I. The ease with which pressure organs can be led into de- 
sired channels on account of their fluidity is applied in the 
operations of casting. Metals which it is desired to make into 
given forms are rendered fluid by heat, and thus converted 
into pressure organs which can readily be run into moulds. 
In similar manner wax, stearine, parafiin, etc., are cast, in 
making candles and the like, the formed material resuming 
its solid state on cooling. Plaster, cement, magnesia or similar 
materials may also be made fluid by mixing with water, and 
then cast into forms -which afterwards become hard by com- 



bination with water and carbonic acid. Other and similar 
methods are adopted for other materials. 

2. Glass, in a plastic state, is formed by pressure in moulds 
or by passage between rollers. 

3. In cases where complete fluidity is unnecessary, the mate- 
rial may be softened by heat, and then shaped in suitable 
presses, the slight fluidity of the material being overcome by 
the mechanical pressure of the machines. 

4. Lead is sufficiently soft to be readily formed by the action 
of a plunger press, and is thus formed into bullets in arsenals, 
and also made into pipe. 

5. The forming of a pressure organ by cooling is shown in 
the action of an ice machine, by means of which water may be 
given the form of sheets, rods, blocks, etc. 

6. Copper, tin, zinc, etc., and also gold and silver are formed 
under the drop press in dies. Steel and wrought iron are 
heated for this purpose ; but sheet steel is stamped cold. 

7. Wire, already considered as a tension organ, may also be 
treated as a pressure organ, having great similarity to a flowing 
stream with its curves. Examples are found in the ingenious 
machines for making hooks and eyes, and also wire chains, 
made by William Prym, of Stolberg. Another illustration is 
the machine of Hoff & Vogt for rolling spiral springs. 

8. Hydraulic or lever presses are used to press clay in a 
plastic condition into various dies to make bricks. Bricks are 
also forms of compressed turf, culm, etc- Chocolate and cocoa 
are also compressed in moulds. 

9. The so-called art work in pressed wood is composed mix- 
ture of sawdust formed into a solid mass by great pressure in. 
suitable moulds. 

10. Papier mache is formed into shape from paper pulp re- 
duced to a doughy consistency, and then subjected to heavy 
pressure. 

11. In the use of moulding machines the pattern is first 
pressed into the moist sand, this being a granular pressure 
organ, this being followed by the casting of the liquid metal in 
the mould thus formed. This gives two applications of form- 
ing, — the first in moulding, the second in casting. 

12. Compresses, such as are used for packing merchandise of 
powdered or fibrous nature, are also examples in point. These 
are used for baling hay, cotton, wool and similar materials 
under very great pressure. 

b. Discharging ; Formation of Jets. 

When a pressure organ is contained in a guiding inclosure, 
and by properly directed pressure is forced out through a suit- 
able mouthpiece, the jet which is emitted is formed with a 
cross section corresponding to that of the mouthpiece used. 
Jets may be formed in this way not only from materials which 
flow readily, but also from those which are of a tough or 
doughy consistency, so that even moderately dense substances 
may be thus formed: 

1. The clay presses made by Schlyckersen and others are 
used to form tiles, drain pipes, etc., by this jet method of form- 
ing, the issuing stream being cut off at regular intervals by a 
wire cutter. The clay in such machines is efi^ectively forced 
through the discharge opening by an arrangement of screw- 
propelling blades. 

2. In the model press the dough is forced by a piston up 
through a die plate in which various shaped holes (such as 
stars, circles and the like) are made, and the issuing streams 
are sliced off by a wire cutter and dried. 

3. The so-called artificial silk of De Chardonne is a jet forma- 
tion of nitro-cellulose. This is made into a semi-fluid mass 
with iron or tin chloride and alcohol, and forced through a 
tube of glass or platinum of about a sixteenth of an inch bore 
drawn to a fine aperture, whence it issues in a hair-like thread. 
It is then toughened by passing through acidulated water, after 
which it is wound on a reel. 

4. In the manulacture of paper the liquid pulp is discharged 
in a broad, flat sheet by its own pressure and the superfluous 
water first removed by absorption, after which the paper is 
dried and made into sheets. 

5. Ivead pipe is made by a process of jet formation in a pipe 
press. The mass, which is only moderately heated, is forced 
by piston pressure through the die in a continuous stream. 

6. The insulating covering of gutta percha is formed upon 
wires used for electrical conductors by a jet action. 

7. The common punching press, used for punching rivet holes 
in plates, really works with a jet action, as has been shown by 
the celebrated researches of Tresca upon the flow of metals. 

8. The drawing press for forming various cups, pans and 
other household articles, also cartridge shells, from sheet metal, 
operates by a kind of jet action, one part of the mouthpiece 
being forced against the other. The powerful presses built by 
Erdmann Kircheis at Aue, and by the Oberhagener Machine 
Works, operate by means of cranks and cams, while those of 



THE CONSTRUCTOR. 



241 



Lorenz, of Carlsruhe, work by hydraulic pressure. Drawing 
presses are much used iu the United States.* 

9. The drawing bench for the manufacture of wire as well as 
rods is an example of jet action. The wire acts both as a ten- 
sion and a pressure organ, since it is pulled through the die in 
which it is formed. Drawn brass tubing is found in a similar 
manner and of various shaped sections. 

10. The manufacture of shot is a variety of jet action, the 
melted alloy of lead and arsenic being poured through a sieve 
and permitted to fall in streams from the top of a shot tower, 
the drops assuming a spherical shape during the fall. 

11. In gas lighting, the shape of the flame is formed by the 
jet tip on the burner, the flat flame in one form being made by 
two round inclined jets impinging against each other. 

c. Internal Flow. 

There are a number of pressure organs which are not homo- 
geneous, being composed of granular and fluid materials, or of 
fluid materials of different densit}'. It is a frequent problem in 
technology to separate such substances so as to divide the 
liquid from the solid, the large from the small, the light from 
the heavy, etc. In general, this can only be done by some 
application of the method of internal flow in the mass of the 
pressure organ. The methods include the use of artificially 
produced high pressures, the natural gravity of the material, 
or in some instances by vibratory or other motion, ;'. e., by the 
action of the living force of the material, or rather by the un- 
equal action of the various portions. The following examples 
will illustrate the various methods : 

1. Presses used for the extractiou of liquids (such as wine 
presses), presses for seed oil, olive oil, also for oil cake, stear- 
ine, beet root, yeast, etc., all act to separate the liquid from the 
solid portion by the action of internal flow. 

2. Filter presses act to separate the fluid from the more slug- 
gish portion of the mass, the liquid passing through the minute 
openings of the filter under the action of the high pressure, 
while the slimy mass remains behind. Filter presses are used 
for separation of colors, stearine, yeast, starch, sugar, potters' 
clay, etc. 

3. The purification of water under natural pressure is effected 
by conducting it through settling and filtering tanks ; also by 
special devices (as that of G. Niemax, of Cologne, German 
patent No. 30,032), by which the water is rendered harder or 
softer, as may be required. t 

4. In mining and machine shop operations, the separation of 
mingled pressure organs by difference of internal flow is effected 
in various ways, showing very effective applications of the laws 
of hydraulics.J 

5. Various applications of sieves are used to separate granular 
materials of different sizes, as are also different devices which 
act by shaking or jigging the material, the separation thus 
being effected by differences of living force. 

6. Centrifugal machines are used for drying yarn, wet clothes, 
etc., although the action iu this case might be more properly 
termed external, rather than internal flow. 

7. Another application of the centrifugal machine is for the 
separation of materials by their difference in specific gravity, 
as in the case of the machines for separating cream frora 
milk. 

8. In the Bessemer process the molten fluid mass of iron is 
penetrated by a gaseous pressure organ, /. e., air, under high 
pressure, producing a violent internal flow and agitation, and 
burning out the carbon of the iron. 

d. Jet Action. 

A considerable amount of living force may be stored up in a 
fluid jet. This may be utilized in a limited number of ways, a 
few of which are here given : 

1. The system of hydraulic gold mining used in California, 
to a great extent, is an important application of the jet. § 

2. Tilghman's sand blast acts by means of a jet of air which 
sets a stream of sand particles in motion. This sand blast is 
used to cut glass, surface metals, sharpen files, clean castings, 
and has many other useful applications. 

3. Machines for cleaning grain are made to throw the grain 
against frictional intercepting surfaces, thus removing dust and 
other impurities. 



*See address by Otierlin Smith (Jour. Frank. Inst., Nov., 1SS6, "Flow ot 
Metals in the Drawing Press"}. The American presses are made for rapjid 
duplicate work, the German for more general service with various special 
attachments, which latter Mr. Smith commends. We may be permitted to 
accept and return the complinieut as regards the other side of the question. 

+ See Z. D. Ingenieure, April, iSS3, p. 377. 

J For an account of the separation of pulverized minerals by means of 
currents of air into portions of various fineness, see Z. D. Ing , April, 1888, 
p. 381. 

§ See Appleton's Cyclopedia of Applied Mechanics, New York, 1880, Vol. 
II., p. 434. 



4. The impinging of a rapidly issuing jet of steam against the 
bell of a whistle causes a series of rapid vibrations, producing 
sound. 

5. In the reed pipes of organs and similar musical instru- 
ments, the notes are produced by the action of a jet of air 
causing the reeds to vibrate. 

6. The "Siren " used for fog signals acts by setting a column 
of air in vibration by rapidly succeeding jets of steam, causing 
a shrill note to be emitted. 

7. In the simple organ pipe a column of air is set into musi- 
cal vibration by a jet of air. The church organ is probably the 
oldest type of pressure organ escapement, the release being 
effected by the hand of the performer. The modern church 
organ consists of a series of the fifth order, namely : a water 
motor (hydraulic escapement), bellows (escapement) and regu- 
lator (checking escapemeiit), stops (escapements), and key- 
board with pneumatic action (escapement). In an organ man- 
ual of 10 octaves there are 120 escapements, each with 11 stops, 
with n different pipe connections arranged together. In this 
connection also may be mentioned barrel organs, iu which the 
closing and releasing of the escapements is effected mechan- 
ically. 

e. Inclosing or Covering. 

As a counterpart to the inclosing of a pressure organ in a 
pipe or vessel, we have the inclosing or covering of a solid 
body by a pressure organ. This occurs when a body is sub- 
merged in a liquid, when its surface at least is covered with 
the liquid. A partial covering may also take place, as, for 
instance, one side of a flat piece, or by distribution after any 
particular plan. These conditions appear in a number of tech- 
nical operations, as will be seen : 

1. In the operations of dyeing, the articles are immersed iu 
a liquid containing the coloring matter, many machines having 
been devised to assist in the operations. 

2. In the various operations of finishing fabrics, heavy flow- 
ing liquids are used, distributed by various brush devices, this 
forming at least a combination of the second order, since the 
liquid must first be distributed to the brushes, and then to the 
fabric. 

3. In coating paper with gum, the gum is distributed in the 
form of a liquid solution. 

4. In the manufacture of colored papers and leathers, the 
color is distributed in liquid form over the desired surfaces. 

5. The various operations of printing from surfaces of stone, 
copper, zinc, steel, etc., involve devices of the third and fourth 
order for the distribution of the printing material before it is 
finally transferred to the paper. 

6. In the operation of printing fabrics and paper hangings, 
the printing surfaces are charged with color by a distributing 
system usually of the third order, and then impressed upon the 
fabric. The printed fabric is dried, usually, by a current of 
warm air, which is merely a gaseous pressure organ. In some 
instances the printed surfaces are dusted with felt dust while 
yet sticky, and then finally dried. 

7. In printing woven fabrics, the processes of (5) and (6) are 
used with a mordant liquid, and the material then immersed iu 
dye, and finally the color washed out of the unprinted portion. 
with water. 

8. The operations of electro plating surfaces with gold, silver, 
copper, brass, zinc, nickel, etc., involve the use of a physical 
apparatus, i. c, the galvanic battery. The disposition of the 
covering may be modified by covering portions with a non- 
conducting material. Another operation iu electrotechnics con- 
sists in the decomposition and deposition of minerals by means 
of electric currents generated mechanically. 

9. In the use of illuminating gas, the method of making a 
gas poor in carbou, and then enriching it either with a rich 
hydrocarbon gas is a form of combination in the line of inclo- 
sing. The incandescent gas lamps operate by the surrounding 
of a network of magnesia or zircon with the flame of a weak 
illuminating gas. 

10. The jet condenser acts by surrounding the discharge ot 
exhaust steam with cold water. 

11. In the surface condenser the tubes are surrounded with 
water and filled with the steam to be condensed, an arrange- 
ment of the second order. 

12. The apparatus for cooling beer consists of an arrange- 
ment of parallel surfaces of sheet metal between which the 
cooling water flows, thus forming an apparatus of the second 
order. 

******** 

The above outline of technological applications of pressure 
organs is only an indication of the systematic treatment of 
which the subject is capable, but cannot here be carried farther, 
as it does not properly belong to the subject of machine de- 
sign. The fifty examples given might each be made the sub- 



242 



THE CONSTRUCTOR. 



ject of a chapter, many of entire books. Even these can lay 
no claim to be a complete survey of the subject ; rather are 
they merely a beginning. They will serve, hovrever, to indi- 
cate at least how great a number of machines and mechanical 
appliances are involved in the use of pressure organs, and ho%v 
these may all be considered to rest upon the same founda- 
tions. 



CHAPTER XXIV. 
CONDUCTORS FOR PRESSURE ORGANS. 

I 334. 

Empirical Formula for the Thickness of Cast Iron 

Pipes. 

Among the various forms of conductors for pressure organs 
mentioned in g 310, the most important are the various kinds 
of pipes. These are made of a great variety of materials, such 
as cast iron, steel, copper, bronze, brass, lead, wood, clay, 
paper, etc. For underground pipes for water, air and gas, cast 
iron has been most extensively used, and it is yet a question 
whether wrought it on or steel will successfully displace it. 

Cyliudrical cast iron pipes, which will first be considered, 
are subjected to so many varying conditions in the course of 
manufacture that the determination of the proper thickness to 
resist moderate internal pressures cannot be made upon strict 
theoretical principles, and recourse must be had to empirical 
formulse. It is customary to subject cast iron pipes to a hy- 
draulic test both at the foundry and also at the place where 
they are to be used, to a pressure of i U to 2 times the working 
pressure to which they are afterwards to be subjected. As a 
protection against rust, the pipes are also coated with asphal- 
tum, applied at a high temperature, or in special cases where 
the expense is permissible, they may be enameled. 

Let the internal diameter be D, and the thickness d, we may 
make : 

For cast iron pipes for water, air or gas. 



= 0.315" + 



D 

So 



(318) 



For cast iron steam pipes and air pump cylinders, 
D 



■■ 0.472/' + 



50 



(319) 



For bored cast iron steam cylinders and pump barrels, 



(5 = 0.787 + : 



(320) 



Example i.— A pipe 12 inches bore, according to (318), should be of 



thickness /5 = o-3i5 ■ 



- - = 0.465", or in practice say J4 in., while for a steam 

80 

pipe, according to (319), we have : [J = 0.472 + — = 0.722", say Y^ in. 

Exam/'le 2.— The pipes used in sections b^ and b.2 of the Frankfort water 
system, already shown in Fig. 955, and subjected to a pressure of 270 pounds 
(18 atmospheres) are 15?^ inches {400 mm.) and 20 inches (508 mm.) diam- 
eter, and according to (318) the corresponding thicknesses should be: 

<J = 0.315 + ^^ = 0. 55 in., and J = 0.31 5 + g^ = 0.565 in. The pipes for the 

water systems of Salzburg, Baraberg, Carlsbad, Goslar and Iserlohn are all 
of thicknesses conforming to formula (31S).* 

Example 3.— The pipes for conveying the compressed air in the construc- 
tion of the Mont Cenis tunnel were 7'^ in. (200 mm.) bore, and subjected to 
a pressure of about 75 pounds, and were exposed in lengths of over 2000 feet 
long during winter and summer. The thickness of these pipes was 0.39 in. 
(10 mm.), and by (31S) the thickness would be 0.41 in. 

Example 4.— The thickness of a locomotive cylinder x^Yx in. diameter 
would be, according to (320): 



787+ ■ 



: 0.944", say I inch. 



The use of cast iron pipes has greatly increased during the 
past twenty years. Manufacturers have been disposed to make 
them of excessive thickness, not onl}^ to obtain the increased 
security, but to add to the cost, — a matter which public ofi&- 
cials are sometimes not disposed to discourage, but which has 
frequently caused such installations to be excessively expen- 
sive. Hundreds of thousands of pounds of metal have thus 
been uselessly buried in the earth, — a waste which the co-oper- 
ation of hydraulic and gas engineers might join in reducing. 



* The foundi-y of Roll, at Solothurn, using a high grade, make still lighter 
pipes, using the formula 5 »; 0.24" + g^* 



335- 



Table of Weights of Cast Iron Pipe. 




Sockets and flauges should be calculated separately. 

?336. 

Pipes for High Pressures. 

In determining the thickness of pipes which are to be sub- 
jected to an unusually high pressure, L,ame's formula (see \ 19) 
may be applied to advantage, i. e. : 



S_ 
D 



■''■(/m~') 



(321) 



in which p is the internal pressure per unit of area, 5" the per- 
missible stress in the material ; the external pressure being so 
small as to be neglected. 

If in the preceding formula we substitute the external diam- 
eter, Z?o = /? -f 2(5, we get : 



D 



I 



S_±p_ 
S-p 



(322) 



This shows that the internal pressure p should in no case ex- 
ceed the permissible stress 6" of the material. If we make 5 
equal to the modulus of rupture for tension, and make p^ S, 
the pipe will be burst according to either formula however 
great the thickness (5 be made. 

For given dimensions and pressures we have for the stress S 
in the walls of a pipe : 



5 


/?o' + J^'' 


I + V' 


P 


D,^ - D' 


I — i/-^ 



(323) 



I? . 



in which the ratio — is indicated by f, as already discussed in 
§ 90. From this we have the following values : 



<!> 


0.50 


0.52 


0.54 


0.56 


0.58 


0.60 


0.62 


0.64 


0.66 


0.68 


0.70 


S 
























p 


1.07 


1-74 


1.82 


1.91 


2.01 


2.13 


2.25 


2.39 


2-54 


2.72 


2.92 


<P 


0.73 


0.74 


0.76 


0-77 


0.78 


0.79 


0.80 


0.81 


0.82 


0.83 


0.84 


s 
























p 


3-15 


3-42 


3-73 


3-91 


4.11 


4-32 


4.56 


4.81 


S." 


5-43 


S.79 


•p 


0.85 


0.86 


0.87 


0.83 


0.89 


0.90 


0,91 


0.92 


0.93 


0.94 


0.95 


p 


6.26 


6.68 


7.23 


7.86 


8.62 


9.52 


10.63 


12.01 


13.80 


16.17 


19.51 



THE CONSTRUCTOR. 



243 



Example I.— In the case of the compressed air pipe of the Mont Cenis 
tunnel, already mentioned, we have 6 = 0.39, D = 7.b75, Dq = 8.66, whence 

^t = yr- = 0.91. For a pressure / = 72 pounds, we have from the above table 

S= 10.63 X 72 = 765 lbs., or, if tested at double the working pressure, the 
metal would be under a stress of 1530 pounds per square inch. 

When p is small, we may use instead of (321) for a sufficient 
approximation (compare Case I., | 19) 

<5 , P , S B 



D '^ S 



and 



2(i 



(324) 



Example 2.- 



ples, we have 5" = 0.5—- 

o 



Applying these formulae to the data of the preceding exam- 

72X 7.S75 _ 



0-39 



: 710 lbs., as an approximate value. 



EjcaiiipU- 3 — A pipe 4 inches diameter is subjected to a water pressure of 
1500 pounds. It is required that the stress ^ shall not exceed 3200 pounds. 

This gives — — 2.13, which from the above table gives i/* = 0.60. From 

/ 

Da . r 

tins we have D^ — = = (>-^^ m- I" some instances the pressure 

0.60 0.60 
may reach as high as 2250 pounds, in which case the stress would reach 
3200 X 1-5 = 4900 lbs. A 4 in. pipe at the Frankfurt Railway Station at 1500 
lbs. has an outside diameter of 6.4 ins. 

Example 4 —The Helfeuberger Water Pressure Engine at Hersbrugg 
near Rheineck, of 40 H. P., has a cast iron feed pipe under a head of 1312 
feet giving a pressure of 569 pounds. The pipe is 14,760 feet long, and 4.6 
ins. diameter, and the thickness in the lower third of its length, is 0.43 ins. 
This gives Dq — 5.46 ins., D = 4.6, and from (333) ■ 



^=569 



5.462 . 



l62 



5.462 _ 4.6'- 

' Exajiipie 5. — Using the empirical formula (31 
sizes under the assumption of 150 lbs. pressure 



= 3357 lbs. 

we have for the following 



Z>= 4" 
5 = 0.36 
y 0.S9 

— 8.62 

P 

S 1293 


6" 

0.39 

0.88 


12" 
0.46 
092 


30" 
068 
0-95 


48" 

0915 

0,96 


7.S6 


12.01 


'9-5 


24.51 


1179 


iSoo 


2925 


3176 



The above values of S are taken as acting tipou the longitu- 
ilinal section of the cylinder, which is the case when a pipe is 
open at both ends. When the ends are closed, there is also to 
be considered the stress on a section at right angles to the axis, 
■which is equal to /.< S. This, combined with the previous 
value, gives for the inclined resultant, \^S'^ -\- {o.^ S)' = \.\2 S 
as the minimum. These conditions exist in the case of a 
cylinder for a hydraulic press. These are usually made of cast 

iron, and the increased thickness 
adds greatly to the weight. It is 
therefore important to use mate- 
rial capable of withstanding a high 
stress, and to take great care in 
construction and in the disposition 
of the material. Repeated melt- 
ings of the iron give more homo- 
geneous castings. Good results are 
also obtained by adding wrought 
iron in the cupola. By thus im- 
proving the quality of the metal, 
the permissible stress may be in- 
creased. A stress as high as 10,000 
lbs. may be permitted when the 
casting is assuredly sound. Simi- 
lar conditions obtain when bronze 
is used. With good bronze, if no 
alteration of form is to occur, the 
stress should not be greater than 
5000 lbs. If it is desired to go 
higher, some harder composition, 
such as manganese bronze, must 
be used. 

A few practical examples will be 
given : 

Example 6.— In raising the tubes of the 
Conwav Bridge, a hj'draulic press of the 
following dimensions was used:* Diam- 
eter of ram, A'= iS inches; bore of cj'liu- 
der, /? = 20 inches; thickness, 5 = S3/j in. 
The load was 650 tons ^ 1,456,000 pounds. 
The pressure in the cylinder being 5900 
lbs. per square inch, we get from (323) 
the stress i" = 10,500 lbs. The cylinder is 
shown in Fig. 1045. 

Example 7. — In the construction of the 
Britannia Tubular Bridge several forms of hydraulic presses were used. 
One of these was a double press with cylinders of the same dimensions 
as in the preceding example. The load on each ram was only 460.5 tons, 
and the cylinder pressure 4190 lbs., giving a stress on the metal 5=7460 
pounds. 

Example 8.— The press which sustained the heaviest load on this great 
work was one which lifted 1144 tons, or 2,562,590 pounds. This was made 




Fig. 1045. 



with a single cylinder with a ram 20 inches diameter, cylinder 22 in. bore, 
and 10 inches thick. The water pressure was 8400, and the stress in the 
metal, according to (323), was 14,500 lbs. ! When the tube of the bridge had 
been raised 24 feet, tlie cylinder gave way, and the load dropped upon the 
safety suppoits, but was seriously damaged. The fracture was not longitu- 
dinal, but in the cross section near the bottom of the cylinder, as shown in 
Fig. 1046. The fracture was doubtless due to the sharp angle at the bottom. 
A new cylinder was made aud successfully used, tlie bottom being altered 
in shape as indicated by the dotted lines. The first cylinder which was 
cast for this press was moulded with the bottom up, but was rejected as 
being porous ; tlie second was cast bottom down, and gave way in use, as 
above described ; the third, for which the iron was melted twice, was suc- 
cessfully used to the end, while a fourth, which was maue as a reserve, was 
not required. 

Example 1:^. — A press designed for making compressed emery wheels has 
the following diniensious : B = 28.35 in., Dq = 40.94 in. K = 27.56 in. 

P= 2,640,000 lbs., from which p = 4425 pounds. We have ih = — -?- = o.6q. 

V 40.94 

whence 5 = 12,134 pounds, wliicli must be considered a high stress. 

More recently the cylinders for hydraulic presses have been 
cast of steel, permitting stresses as high as 20,000 to 28,000 
pounds. Modifications in the method of construction may 
also be made to enable cast 
iron to stand higher pressures. 

The danger due to casting the 
bottom in one with the cylinder 
may be avoided. The method 
used by Hummel, of Berlin, is 
to make the cylinder as a ring, 
aud the bottom as a separate 
plate (Fig. 1047). 

Lorenz, of Carlsruhe, makes 
the bottom separately and screws 
it in, as shown in Fig. 1048. 

By increasing the diameter of 
the ram to exert a given force, 
the pressure of the water re- 
quired will be reduced, aud the 
stress 5 will be less. This is not 
attended with a proportional in- 
crease in the amount of metal 
required, but on the contrary 
with a reduction. If the cross 
section of the cylinder be F, we 
have F^=TT [D -\- 6) 'S. Substituting the value of i from (321), 

nZ) - 2h 
we get P = % -^ and introducing K, 







Fig. 1046. 



2.P 



^= t; s~--p (335) 

which, for any choseu value of 6", diminishes as/> is reduced. 

Exajitple lo. — In a hydraulic press by Hummel, for making rollers of 
compressed paper, there are two cylinders of the form shown in Fig. 1047, 
placed side by side. The diameter A' of the ram is 23 inches, and the cylin- 
der diameter D 24 inches, the thickness 6 being 8J-2 inches. The load P on 
the ram is 2,200,000 pounds. The water pressure is 5174 pounds, and the 
stress on the material about 10,000 pounds. If we increase K to 26 inches, 
we have, since this is % the preceding value, the value of / reduced to {%)- = 
0.79 of the previous amount, or 4087 lbs. If we now make the relation be- 
tween the inside and outside diameters of the cylinder the same as before, 
we have the same relation between .5" and /, hence >S= 10,000 X o 79= 7,900 

K 

find the relation between the cross sections of the two cylinders will be also 
as 0.79 to I. Hence this alteration in dimensions which reduces the pres- 
sure in the cylinder also causes a reduction of about 20 per cent, in the 
amount of material. 

I 337- 
Wrought Iron and Steei- Pipes. 

Wrought iron pipe is in very extensive use for conveying 
gas, water, air, petroleum, as well as steam. These pipes are 
made either by the process of welding during passage between 
rollers or are riveted while cold. The former method produces 
either a butt- or lap-welded joint, the seam being parallel to 
the axis of the pipe, and more recently pipe has been made in 
America with a spiral lap-welded seam.f After welding the 
outside of wrought pipe is generally made smooth by passing 
betv.'een another set of rolls after re-heating, whence it is 
sometimes called "drawn" pipe. Pipe is also made of mild 
steel in the same manner as if wrought iron. The Mannes- 
mann system is also used for rolling tubing from the solid rod 
of steel, copper, delta metal, etc., the product being without 
any seam. 

Welded tubing possesses a great resistance to external pres- 
sure and to tension, but a less resistance to iuternal pressure. 
Butt-welded pipe should not be subjected to a greater stress 
than 5^1 500 lbs. ; but for lap-welded pipe 6" may reach Sooo 



*See Clark, The Britannia and Conway Tubular Bridges. London, 1850. 



t See Engineering and Mining Journal, April 7 and 14, 18 
tific American, June, 1SS8, p. 377. 



also Scicn- 



244 



THE CONSTRUCTOR. 



to 12,000 pounds. Spiral lap-welded tubing has been tested 
to pressures corresponding to stresses from 30,000 to 40,000 
pounds, according to the quality of the material used ; but in 
practical service lower values are used. The Mannesmanu 
tubes have been used without deformation almost to the elastic 
limit of the material, which, with cast steel and with Siemen's 
open-hearth steel, reaches 25,000 to 50,000 pounds, and there- 
fore possess a utility to which welded tubes have not attained. 



-!K 




V X-0—i.i^ 




Fig. 1047. 

Example i — In the oil pipe line showu in Fig. 954, 6 inch lap-welded pipe 
is used, I'i; in. thick, at a pressure of about 1000 pounds. We get from (324) 

„ 6 X 1000 

■J = —r^ = 9600 lbs. From (323) we have more accurately 5 = 1000 

(6.625)° -f (6)= 



(6.625)=- (6)2 



98S7 lb.=. 



Example 2.— It a spiral lap welded pipe had been used for the preceding 
example, the thickness 5 need only have been ^V in. 

Example 3.— If a Mannesmaun tube of Siemens steel had been used for 
the high pressure water service of Example 3, \ 3 j6, and the stress put at 
the moderate limit of 15,000 pounds, we get from' (324) Z? ^ 4",/ = 1500 lbs., 

1500 X 4 
^ ^ ■ = 0.2", and from {323) we get more accurately 



i5,o°o X 2 



.S= 1500 



(4.4>' -f (4)' 
(4.4)=- (4)- ■ 



The steel pipe would weigh onlv about J that of the corresponding cast 
iron pipe.l 



As an example of the efficiency of this construction, Mr. 
Hamilton Smyth cites an installation over two miles long and 
under a head of 550 ft., the pipe lying on the surface of the 
ground and only protected from changes of temperature by a 
roof of roughly nailed boards, and in which the total loss by 
leakage was only 3 to 4 cubic feet per minute. 

As a consequence of the successful use of these pipes for 
mining purposes, they were next used for more permanent 
service as for water supply of cities, and with excellent results. 
Two such pipes were put in for the supply of drinking water 
for San J'"rancisco, and a third pipe, many miles long, was sub- 
sequently added. For large diameters in permanent installa- 
tions the sections should be riveted together, while for smaller 
diameters the joint may be made with lead, as hereafter de- 
scribed. The following table will illustrate some important 
constructions of this kind. 



Cherokee . . . 

Virginia City < 

Texas Brook . 
Humbug . , . 



ft. 
1870I 12,800 



1S7: 
1873 



37,116 
37,116 

4,44° 
1,194 



u 








s 
■p 


^ 


in. 


ft. 


3° 


8S7 


II 


1722 


10 


1722 


17 


781 


2b 


120 



Description of Pipe. 



lb. 

17,500 Sheet Iron, double riveted. 
i4,ooo| " " " " 

14,000 Lap welded, screw connections. 
17,000 Sheet Iron, double riveted. 
ii,5oo|^" sheet iron, single riveted. 



Also may be noted the Kimberley water works in England, 
14 inches diameter, % in. thick and eighteen miles long. The 
superior economy of wrought pipe over that of cast iron is 
worthy of greater attention. 

lu order to illustrate the arrangement more fully of an in- 
stallation of such pipe the inverted siphon in the vallej' of the 
Texas Brook, constructed by Mr. Hamilton Smyth, is given, 
Fig. 1049. The difference in the level is 303.6 feet, and the total 
length 4438.7 ft. The pipe is in lengths of 20 feet and the 
figures in the diagram indicate the gauge thickness of the sheet 
iron in the various portions. 

The average diameter of the pipe is 17 inches and the highest 
value of the stres .S was calculated as equal to 16,500 pounds ; 
some of the plates were too thin and the stress in such 
places reached 18,000 pounds. The inlet is so shapted that the 
coefficient of contraction reaches 0.92. The pipe is bedded in 
gravel 12 to iS inches deep, and passes entirely under the bed 




Fig. 1049. 



Riveted pipe of wrought iron have been successfully used iu 
America for conducting over long distances, and valuable in- 
formation has been furnished by Mr. Hamilton Smyth, Jr., 
upon this subject.* 

Wrought iron riveted pipes were first used in California, 
made steel metal yV in. thick, to take the place of the canvas 
hose then extensively used in the operations of hydraulic rain- 
ing. The pipes were made of ordinary sheet iron, there being a 
single row of rivets, driven cold, and the joints made simply 
by inserting the end of one section into the next, as in the case 
of stove pipes. These first attempts succeeded Ijeyond all ex- 
pectations and were followed by numerous installations, in sizes 
reaching as high as 22" to 30'' diameter and sections 18 to 25 
feet long. ^ A satisfactory protection against rust was obtained 
by immersing the finished pipes for a few minutes iu a boiling 
mixture of asphaltum and tar. If the fit of the ends was too 
loose to make a good joint the smaller pipe was wrapped with 
tarred cord, leaky places being stopped w'th wedges of wood 
and the small leaks being checked by sawdust admitted with 
the water. 



of the stream. During a large part of the year the siphon is 
not full of water and hence entraps much air. In order to per- 
mit this to escape, air valves of the coustructiou shown in Fig. 
1050, are attached at suitable points, fourteen iu all being used. 




* See Engineering and Mining Journal, May and Tune, 1884 : also Journal 
of the Iron and Steel Institute, 1886, No. I, p. 133. 



Fig. 1050. 

These are simply heavy cast iron flap valves with rubber ring 
packing. When the chamber is filled with air the valve falls 
open by its weight but is closed by the action of the water 



THE CONSTRUCTOR. 



245 



when the air has escaped. In case of a rupture in the lower 
portion of the pipe, the air valves in the upper portion prevent 
the collapse of the pipe from atmospheric pressure. 

I 33S. 
STEAM Pipes. 
When steam is to be conducted to considerable distances, the 
condensation which is due to loss of heat through the walls of 
the pipes becomes so great that it is necessary to surround the 
pipes with a non-conducting covering. Materials for covering 
steam pipes play quite an important part in the science of 
steam economy and their manufacture constitutes an extensive 
industry. The importance of this subject has long been appre- 
ciated, having been considered, among others by the Industrial 
Society of Mulhouse more than sixty years ago. In these in- 
vestigations the measure of effect is the amount of water con- 
densed by a unit of surface, as one square metre per second. 
The following table will indicate some of the results obtained.* 



Material. 



Material of Covoriug. 



Uncovered Pipe 
Pimont's Mass. . 
Straw 



Graninies 
condensed 

per sq 

metre per 

second. 



2.84 gr. 
1.56 " 
0.9S " 



Material of Covering. 



Clay Pipe . . 
Cotton Waste 
Felt 



GraniTOCs 
condensed 

per sq. 
metre per 

second. 



i.i2gr. 

1-39 " 
1-35 " 



The so-called Pimont's Mass, consists of loam and cows' hair, 
60 mm. (2^ in.) thick. The straw was first laid on longitudi- 
nally 14 mm. (fj in.) thick, and then wrapped with straw 15 
mm. (J's in.) thick. The cotton waste was 25 mm. (i in.) thick 
covered with canvas. The felt was saturated with rubber. 
Straw shows the best results, the condensation being only one- 
third that given by the uncovered pipe. 

These experiments have not great present value, partly be- 
cause the comparison by condensation of water is not altogether 
reliable, and partly because new material for covering pipes 
have since come into use. The Society of German Eugiiieers 
( Verein Deutscher Ingenicin'c) has undertaken a series of ex- 
periments from which results of value are to be anticipated. 
In the United States, Prof Ordway, of Boston, has made some 
very beautiful investigations, the results being in two series, 
the first by the method of measuring the condensed water, the 
second by the calorimetric method.! The unsatisfactory char- 
acter of the method of condensation is apparent, as it was 
found, for example, that a portion of pipe 2 feet long condensed 
32S grammes of water per square foot per hour, while 30 feet of 
pipe gave only 140 grammes per square foot per hour. It is 
also to be noted that Prof Ordway's first researches showed 
much less condensation for the uncovered pipe than appeared 
in the Mulhouse experiments, so that no definite conclusions 
could be deduced. The calorimetric method appears to be 
much more reliable, as the results appear to be more consistent. 
From the great number of experiments the two following tables 
have been selected. 

Table I. 



Air Space 

Carded Cotton .... 

Feathers 

Wool • • • 

Calcined Magnesia . . 
Cork Charcoal, coarse . 
Calcined Magnesia . . 

Wool 

Lampblack 

Carbonate of Magnesia 

Fossil Meal 

Wool 

Asbestos 

Zinc, White 

Fossil Meal 

Pine Charcoal 

Carbonate of Magnesia 

Hair Felt 

Lampblack ..... . 

Chalk 

Graphite 

Calcined Magnesia . . 

Zinc, White 

Pumice Stone 



Per Cent. 


Ivilo-Cent. 


Solid Material. 


Heat Units. 


0.0 


1302 


I.O 


310 


2-0 


321 


2.1 


301 


2.3 


335 


31 


343 


4.9 


340 


5-6 


220 


5-6 


266 


6.0 


371 


6.0 


393 


7-9 


238 


8.1 


'329 


8.8 


466 


II. 2 


426 


11.9 


376 


15.0 


416 


1S.5 


177 


24.4 


286 


25-3 


560 


26.1 


1922 


28.5 


1156 


323 


1164 


,34.2 


Hi 



Plaster of Paris 
Common Salt . - 
Anthracite Coal , 
Fine Sand . . 
Coarse Sand . 



Per Cent. 


Kilo-Cent. 


Solid Material. 


Heat Units. 


36.S 


S39 


48.0 


I9S3 


50.6 


968 


51.4 


i6go 


52.9 


1684 



Temperature of steam 155° C. All coverings i inch thick 
= 25.4 mm. 

This table gives noteworthy, and in many cases unexpected 
results. It is important to note that in all cases the trans- 
mission of heat bears a definite relation to the percentage of 
solid matter. For instance, calcined magnesia gives off 335 to 
1 1 56 heirt units when the percentage of solid matter ranges 
from 2.3 to 28.5. Asbestos makes an unfavorable showing, and 
lampblack gives good results but is inconvenient to use ; wool, 
is excellent. In practice the cost is of course an important 
consideration. 

Table IL 
Temperature of steam 155° C. 



Glazed Cotton Wadding 



Wool Wadding , 

Calcined Magnesia, loose . . 
" ■" crowded . 

" " compressed 

Carbonate of Magnesia, loose 

" " crowded 

" " compressed 

Fossil Meal, loose 

" " crowded 

j Cork in Strips 

'i Silicated Cork Chips 

Paste of Fossil Meal and Flair . 

Carded Cotton 

Rice Chaff, straw board .... 



rhickness. 
Milli- 
metres. 


Per Cent. 

Solid 
Matter. 


Kilo-Cent. 
Heat units. 


50 


I-OJ 


I29.I 


40 


1-3 


193-4 


3° 


1-7 


205.5 


20 


2.5 


326.4 


15 


3-4 


424.2 


10 


5-1 


502.4 


25 


5-6 


219.8 


25 


2.3 


335.2 


25 


4.9 


340.1 


25 


28.5 


"55.9 


25 


6.0 


370-9 


25 


9-4 


386.7 


25 


15 


416.5 


25 


6.0 


393-4 


25 


n.2 


435-8 


15 


? 


87.1 


30 


? 


59-2 


9 


1.0 


69.4 


50 


? 


>57-7 


12 


? 


71.9 



This table gives a comparison of fibrous and granular ma- 
terials. In the first cases the same material was successfully 
compressed, reducing the thickness and increasing the density, 
showing and increasitig loss of heat. Ordway recommends 
cork as the best material, especiall}' in the form of cemented 
chips, which may be formed into semi-cylindrical sections, as 
has already been done in Germany.]] 

Ordway does not advise air space under the covering, but 
rather recommends the filling such space with alight powder. 
Of all the materials tried he recommends in the order given: 
Hair Felt, Cork, Fossil Meal, Magnesia, Charcoal and Rice 
Chaff.f Prof Ordway remarks that " it is useless to make the 
testing apparatus of cumbrous dimensions, for as in chemical 
analysis we use a gramme or less of the sample, instead of kilo- 
grammes, so in physical experiments increase of size does not 
uecessaril3' enhance the accuracy of the results." 

In long stretches of steam pipe the expansion from the heat 
demands the use of some compensatory device or expansion 
joint.** 





Fig. 1 05 1. 

Some of the forms in general use are shown in Fig. 1051. As 
a, is a packed expansion joint; h, is a bent copper pipe; c, a 
drum with flexible steel diaphragms. 



* This table has been kept in the metric system, as it is only available for 
comparison. — Trans. 
\ See Trans, Am Soc. Mechanical Engineers, Vol V. p. 73 ; Vol. VI. p. 168. 



t The cork was put on like barrel staves, with a slight air space beneath. 

^ The cork was chopped into small chips and mixed with ^Y^ t;mes ita 
weight of water glass at 30° Beaume. 

II See Z. D. Ingenieure, 18S6, p. 38. 

\ A new, and efficient as well as cheap material made of common flour 
paste and saw dust, is described in the Revue Industrielle, Sept. 1S8S, p. 346. 

** This "compensation " does not neutralize the expansion, as in a pendu- 
lum, but only renders it harmless. 



24^ 



THE CONSTRUCTOR. 



Fig. 1052 shows a U joint with packed connections. The 
forms given in Fig. 105 1 generally require one position of the 
pipe to be held fast ; that in Fig. 1052 permits both lengths of 
pipe to remain free. 




Fig. 1052. 

In calculations the actual amount of expansion due to any- 
given temperature we may put the expansion, if t, be the dif- 
ference of temperature in degrees for : 

Materia!. 
Cast Iron 



Wrought Iron 
Copper . . . , 



Brass 



Centigrade. 


Fahrenheit 


t 


t 


90, 1 00 


162,180 


t 


/ 


~^476oo 


155,280 


t 


t 


58,206 


104,760 


t 


t 



53,500 



92,30 



Example. A cast iron pipe 98.4 ft. long, (= iiSi.i inches). At a tempera- 
ture of 50° F. is filled with steam at 63 pounds pressure, =310° F. The ex- 
pansion will then be 

J181.1 X 260 

; — .; = T.Sg.in. 

162,180 ^' 

I 339. 

Pipes of Copper axd other Metal. 

Brazed pipes of copper when used as conductors of steam, 
should not be subjected to higher stresses than 1500 to 2000 
pounds, since the brazed joint is not reliable and reduces the 
strength of the cross section of the metal about one-third. The 
heat due to the temperature of steam at pressures from 60 to 
100 pounds also reduces the strength of the copper from 10 to 
12 per ceut.* 

Seamless pipes made from the solid metal, or rolled by the 
Mannesmann process, can stand stresses from 8,000 to 10,000 
pounds, and when made by forcing in the hydraulic press (see 
? 333, b- 5) only a stress of about 700 to Soo pounds. 

Wooden pipes for water conductors, made water-tight with 
cement, have been made by Herzog of Logelbach with excel- 
lent results; the most recent being 71 in. diameter, and 5900 
feet long. 

Pipes made of paper coated with asphalt have been used to 
a limited extent, but do not stand the heat of the sun. 

§ 340. 

Resistance to Flow in Pipes. 

The resistances which oppose the motion of a liquid in a pipe 
are due either to changes in the direction of motion, to changes 
in the rate of motion, or to the resistance of friction. We can 
only here consider a few cases, and those will be limited to the 
flow through pipes. 




Frictional Resistance. — When a flow of water takes place in a 
vessel with flat walls, through a cylindrical tube, Fi.g. 1053, the 
difference of level between the surface of the water and the 



* Investigations made after the explosions on the Elbe and the Lahn will 
be found in Engineering for August, 18S8, pp. 113, 116, 125. These gave for 
the modulus of rupture jT for tension; for hard brazed pipes A' = 33,400 ; 
for seamless electrically deposited pipes, K = 50,000. The reduction in 
strength due to heat is given according to the old but reliable experiment 
of the Franklin Institute. 



mouth of the discharge pipe being /;, we have, according to 
Weisbach : 



// 



O+co+4) 



(326) 



in which / is the length and d the diameter of the tube in feet, 
and V the velocity in feet per second. The volume of flow 
will be : 



Q-- 



•d'v . 



(327) 



per second. 

In (326) fg is the coefiicient of friction for the orifice of influx, 
and C the coefficient of friction for the rest of the tube. 

The coefficient Cq, when the entrance is a sharp angle, be- 
comes considerable, having a mean value of 0.505, but when the 
entrance is carefully rounded it may fall as low as 0.08. In the 
latter case, for long tubes, Co may be neglected. f 

I'or the coefficient of friction C in the pipe various deductions 
have been made. The conditions which affect the flow of water 
in pipes are numerous and variable. In cylindrical pipes the 
particles arrange themselves in such a manner that those in the 
axis move with the greatest velocity, and each successive annu- 
lar sheet moving slower, while the particles in contact with the 
walls of the tube remain practically at rest, so that the velocity 
of each annular film, from the wall to the axis is a function a 
the distance from the wall to the centre, increasing from zei 
to the maximum. 

In the case of gases the velocity of adjacent rings appro*, 
mates much more closely than with liquids. In both instances 
the resistance is the sum of the friction of the successive annu- 
lar layers upon each other. 

In practice, the variation of the section of the pipe from the 
circular form must be taken into account, and also the rough- 
ness of the walls. The mathematical expression of these rela- 
tions cannot be a simple one. In practice also many disturbing 
influences exist, such, for instance, as ice, weeds, etc. In all 
comparisons with calculated resistances it is therefore essential 
that the walls of the pipe should be ascertained to be smooth 
and clean. 

The Societ)' of German Architects and Engineers has in 
progress modern iuvestigations conducted by several of its 
members with a view of determining the most useful formula 
for finding the value of C for water. The value which such a 
formula would possess is undoubted, but before it can be satis- 
factorily determined the fundamental principles of the subject 
must be determined.! We are at present obliged to use for- 
mulas previously determined. Among these the formulas of 
Weisbach and of Darcy are most available. If we express the 
loss of head in the height // by friction //j, in feet, we have for 
water, according to Weisbach : 



h,. 



I 



0.01439 -I- / — 



(32S) 



all dimensions being expressed in feet, and "• being the accelera- 
tion of gravity.^ We have for: 



V = 0.1 
C = 0.06S6 



0.0527 



0-3 
0.0457 



0.4 
0.0415 



0.03S7 



11 = 0.5 
C = 0.0365 


0.7 
0.0349 




o.S 
0.0336 


0.9 
0.032s 


C = 0.0315 


C.0297 


0.02S4 


2 
0.0265 


3 
0.0243 


C = 0.0230 


6 
0.0214 


8 
0.0205 


12 
0.0193 


20 
0.0182 



According to Darcy we have for water : 

I v'^ f ^ , 0.00166 ^, / il^ 

— — == 1 0.019S9 -^ — j — — . 

d 2o- \ -^ d J d 2g 

which gives somewhat .greater values thau does Weisbach 
formula for the higher velocities. 11 



}u = ^ 



(329) 



t If the tube starts from another tube instead of from the side of a reser- 
voir, the coefficient of resistance becomes much greater and much care must 
be given to the shape of the entrance. See Hertel, Zeitschr. D. Ingenieure. 
1SS5, p. 660, also \V. Roux, Jenaische Zeitschr. fur Naturwisseuschaften, 
Vol. XII., 187S. 

X See a Memoir of tlie D. Arch- u lug.-Yereine, edited b^' Otto Iben, pub- 
hshed by Meissner, Hamburg, iSSo. This must be used with caution oa 
account of numerous typographical errors. 

g This and the two following formulas may also be used when in addition 
to the height /ji a second height h-^' is to be added due to the contraction ot 
discharge. This is only of importance in the case of high pressure water 
transmission, and experimental researches are to be dej:'*ed. 

il Recherches Experimentales, etc. Paris, Mallet-Bachelier, 1857, 



THE CONSTRUCTOR. 



247 



The formula of M. de Saint Venant, which gives lower results 
than either of the above, is : 



Aj = (0.0321 V ' 



2 / v^ 
^ d 2g 



(330) 



If we insert in equation (327) the value for v from the equa- 



tion /?! 



'D 



— we get : 

2g 



:=r-)^^..4.f=(^)f.4=-^.^^- 



2g 

By assuming f constant, as proposed by Dupuit, * we may- 
state the practical formula : 

'h 



Q" 



Cd^- 



Dupuit makes C = 0.03025649, whence C becomes 1313, and 
we have : 



rfS; 



(331) 



L(^\ 

'H V 36.237 

and hence for an approximate formula : 

•^-^WS) '»" 

These formulas cau be applied so that first from the given 
values of Q and / aud the friction loss of head //p the diameter 
D may be determined, and then by making Z> somewhat larger, 
and applying the formula of Weisbach or Darcy, the excess of 
head over friction determiued. This will be illustrated by a few 
examples : 



Example i — The large inverted siphon described in ; 
O = 17" = 1.4-7 f'-i ' = 4438 ft. and i? = 32 cu. ft. 
32 



337. Fig- 1049. gives 
From th.s we get 



— „ , , — , = 21.2 ft. per second. 

o. 7854 X (1.417)- 
These ^ve in Weisbach's formula : 

0.017155 



= 0.01439 + 



%/- 



: 0.0181, 



and from (328) h-^ = 395.6 ft. 

The actual difference of level is 303.6 feet, and hence the coefficient as de- 
termined from Weisbach, is too high. The coefficient determined from the 
given difference of level is ^ = 0.0155, and as a flow takes place the actual 
coefficient must be somewhat less. According to Saint Venant's formula 
(33°) ^'1 = 293 feet, which is slightly under the actual difference of level. 

Example 2. — In the work of Iben, already referred to, is given a case in 
Stuttgart in which /= 3614 ft., I) = 0.33 ft , v = 2.063 ft. From these data, 
Weisbach's formula gives ^ — 0.0263, and thence k-y = 19 ft. The actual value 
is 23.2 ft., which corresponds to ^ = 0.0332. The difference is probably due 
to the construction, there being two stop valves and six elbows included in 
the resistance. 

Example 3. — Another instance in Stuttgart is as follows : / = 302 ft. ; 
D = 0843 ft ; !J = 5.897 ft. 

The friction head as determined by observation for / = 328 ft., was ^2.89 ft. 
According to Darcy, the friction head would be 76.57, which is quite close to 
the e.'cperimental resuUs. In other instances, however, Darcy's formula 
has not agreed so well with experiment. 

When air is used instead of water, Weisbach gives for the 
height of a column of water equal to the frictional resistance : 

/'i = fi — r = o-°25 -7- ■ .... (333) 

a 2gE a 2g t 

in which e is the ratio of the density of the air in the pipe to 
that of the external atmosphere. Since is e always greater than 
unity when the air in the pipe is under pressure, //j is smaller 
than is the case for water, especially when the pressure of the 
air is great. Valuable experiments upon the transmission of 
compressed air have been made by Engineer Stockalper at the 
St. Gothard tunnel. f These showed that Darcy's formula (329) 
served well for air when the results are multiplied by the ratio 
of the density of the air to water. Professor Uuwin has given 
some valuable researches upon the friction of air, in which he 
shows the important influence which D exerts upou C.J 

Exa7nph 4. — At the construction of the Hoosac Tunnel it was observed that 
the pressure of compressed air fell from 821 pounds per square inch to 801 
pounds in being transmitted a distance of about 118,000 feet. 

Resistance in Angles and Bends. — The resistance due to an 
angle, such as Fig. 1054 a is important, and is dependent upou 
what Weisbach calls the semi-angle of deviation, /3, according 
to the following formula : 

7;J 



K-=L 



V' 



(°-9457 ■S'Z'J' /5 + 2-047 sin^ (3) — . 



(334) 



*See Dupuit, Traite theoretique et pratique de la condxute et de la dis- 
tribution des eaux. Paris, Dunod, init ed. 1854; snie ed. 1865. 

f Stockalper, Experiences, fates au Tunnel de Saint Gothard, sur I'ecoule- 
ment de I'air comprime. Geneve, 1879. 

t The coefficient of friction of air flowing in long pipes. Proc. Inst. C. E). 
tyOndon, 18S0. 



from which we get : 

^ = 10 20 30 40 45 50 60 70 

C2 = 0.046 0.139 0.364 0.740 0.9S5 r.260 1.861 2.431 

Example 5.— In a right angle beud (3 = 45°, the loss is practically equal 



to- 




FlG. 1054. 

In the case of bends, Fig. 1054 b, the resistance is not so great, 
but is too large to be neglected since we have : 

'■ '90 ■ 2g' 

The ratio of the radius of the tube to the radius of the curva- 
ture of the bend affects the coefficient as below : 

0.5 Z? 



(335) 



-^ = O.I 

r 


0.2 


0-3 


0.4 


0-5 


C2 = 0.131 


0.138 


0.158 


0.206 


0.294 


^— = 0.6 
r 


0.7 


0.8 


°9 


I.O 


fj = 0.440 


0.661 


0977 


1.408 


1.97S 



Exaj/tple 6. — For a right angle bend in which r ^ D we have : 

,, 45 »- "3 

h^= 0.294 = o- 147 

90 -zg ^' 2g 

or only about f the resistance of a sharp bend with any curvature. 

Resistances due to Sudden Changes of Cross-Section. — When 
water which is moving at a velocity v^ suddenly changes to 
another velocity v, see Fig. 1055 a, it experiences a loss of 
pressure which, according to Weisbach, is equivalent to a height : 



v^—v^ _ Z' F^ ^ ^ ^ — r ^ 



(336) 



F and F^ being the respective cross sections ; also Fzi^= ^\T\- 



Doubling the cross section causes a loss of head equal to 



^S 




Fig. 1055. 

For gate valves. Fig. 1055 b, or cocks. Fig. 1055 c, there is a 
loss due to the amount of contraction. For gate valves we 
have from Weisbach : 

Openings = % j{ 3/i K f s 3/ ^ 

F 

~ = 0.159 0.315 0.466 0.609 0.740 0.836 0.94S 

F 

C3 ^ 97. S 17.00 5.52 2.06 oSi 0.26 0.07 

and for cocks : 



20" 



40" 



50° 



60° 



65° 825 



10.850 0.692 0.535 0.3S5 0.250 0.137 0.091 



Angle ■ 

^1 
F 

C = 0.29 1.56 5.47 17.3 52.6 206 4S6 CO 

From the above tables it will be seen how important an in- 
fluence is exerted by valve chests, mud traps aud the like upon 
the flow of water. In all such cases it is important to modify 
the suddenness of the change of velocity by rounding and curv- 
ing all angles in the passages, and in this way a large part of 
the loss may be obviated. For gaseous fluids the resistance is 
less, but is at the same time sufficiently important to be care- 
fully considered. For a fuller discussion of the resistances 
offered to water in canals and streams the reader must be re- 
ferred to special treatises on the subject. 



24 « 



THE CONSTRUCTOR. 



?34i. 

Methods op Connecting Cast Iron Pipes. 

One of the most frequently used methods of connecting cast 
pipes is by means of the common flange joint, Fig. 1056. 






Fig. 1056. 

The proportions are given in the illustration. Formerly it 
■was customary to raise a small bearing surface inside the boit 
circle, but this is generally omitted now, and the entire surface 
of the flauges finished, making a much better joint, although 
a trifle more expeusive. In many instances a ring of copper 
wire, let into a groove, is used to make the joint. For pipes 
which are not subjected to very high pressures the number of 
bolts A, is determined from the following : 



^ =2 + 



D 



(337) 



in which D is the diameter of the pipe inches. This would give 
for a pipe 4 inches in diameter four bolts, and for one 3 inches 
diameter 6 bolts. According to (337) an air pump cylinder 60 
inches in diameter would have 2 -f- -'i- = 32 bolts. 

When the pressure is known to be great, or for cylinder lids, 
etc., the following formula is to be preferred : 



2400 



© 



(338) 



in which d is the diameter of the bolts, D the diameter of the 
cylinder, and a the pressure in pounds per square inch. This 
assumes the diameter of the bolt at the bottom of the thread to 
be o.S d, and the stress in the bolts to be 3500 lbs- as in for- 
mula (72). 

Example. — .\ steam cylinder 40 inches in diameter, subjected to a pressure 
of 60 pounds, would have according to {320) a thickness of 5 = 0.7S7 + f^f, = 
•L^z in. This gives from Fig. 1056 for the bolts, ^ = ^ X i?c = 1.5S, say \^^ in., 
and these values in (33S} give for the number of bolts : 



."I = 



60 



2400 \ 1.58 
(Compare close of Chapter XXVI). 



i)" 



16 bolts. 




Fig. 1057. 

Flanges with ears, as shown in Fig. 1057, are frequently used, 
the thickness being made 2 to 2.5 &, instead of 1.6 t!, on account 
of the smaller flanges. 

On the Prussian State railways flange joints are made with a 
lenticular shaped ring inserted in the joint, as shown in Fig. 
1058. 

This permits a certain amount of motion and gives good re- 
sults in practice. The following table of dimensions is based 
on one used on the Prussian railways : 



D 


2 


= K 


2^ 


^H 


3 


3H 


4 


4K 


s 


sH 


6 


r>, 


3 


3K 


■IM 


■■?« 


4 


3">l 


5 


'.H 


6M 


6% 


iM 


r 


= '/« 


^/4 


2H 


^'A 


3 


iH 


4 


AM 


4H 


A'4 


s 


}4 




t-E 


A 


H 


H 


H 


Ji 


Ji 


iS 



Fig. 1059 a shows a cast iron bend with flange. The bend 
should not be too sharp, in order to avoid excessive resistance 
to the ilow of the water. ^See Example 6, I 340.) Bends of 
this sort require a separate pattern to be made for every different 



angle. Brown's joint is more convenient in this respect, Fig. 
1059 b. The bolt holes in this form should be drilled in only 
one of the flanges first, and the other flange marked off in 




JQ — 



place. For any flange angle c< the pipes may be connected for 
any angle between 2 c/ and iSo°. In the illustration c< = 40°, 
which answers for most practical purposes. 




Fig. 1059. 



Bell or socket connections are much used for gas and water 
pipe. The joint is caulked with lead, which may conveniently 
be made in ha;lf rings and driven in, or run in in place, a pack- 
ing of oakum being first driven in. 




Fig. 1060. 

The large end of the pipe is called the bell, the other the 
spigot. The dimensions of the various parts in Fig. 1060 may 
be taken as follows, the thickness d being determined from for- 
mula (318), i.e., i = 0.315 



80 

Thickness of bell, i^ 

Thickness of bead, k 

Inside length of bell, /, 

Length of bell reinforcement, /.^ 

Outside length of bell, / 

Space for packing, b 

Depth of lead ring // 

Length of bead on spigot. a 

Thickness of bead, c 



=•0.375'' + 0.0135 Z?. 
= 0.7" + 0.0025 D. 
= 2.625'' + 0.1 1 D. 
= 2'' 4- 0.09/?. 
=^ 4.625" + 0.20 Z). 
= 0.1875" + 0.007 .0. 
= 1.125" -f 0.07 £>. 
= 1.2 6. 
= 6 -\- b — 0.0625". 



Some makers put a bead around the inside edge of the bell 
to assist in retaining the lead packing, but others consider this 
but little use, owing to the softness of the metal. More recently 
the bead has been altogether omitted from the spigot end, a 
shoulder being cast on the inside of the bell instead. 

In Belgium a joint is used in which a gum ring of globoid 
form (see Fig. 637 a) is used instead of the lead packing, the 
ring rolling in as the spigot is pushed into the bell. 

Fig. 1061 is Petit's pipe joint. A gum ring is inserted in the 
short bell, and one clamp being connected the pipe is used as a 
lever to compress the gum ring, the second clamp can be 
secured. This coupling, which was used in the extensive water 
system of the camp at Chalons, is cheap and can be rapidly 



THE CONSTRUCTOR. 



249 



connected, and possesses a certain flexibility which permits it 
to be used in running a line of pipe over uneven ground. 





""Hf'^ 



Fig. 1061. 

A form of screw connection for cast iron pipe is shown in 
Fig. 1062. The screw thread is cast on the pipe and a leaden 



^-- 




•^asket is placed so as to pack the joint outside of the screw 
connection.* This may be considered as a flange joint with a 
single central bolt, which latter is made large enough to permit 
the pipe opening to pass through it (see \ 86). Since the pipe 
must be revolved in making the connection, it is necessary to 
provide wrenches of suitable size for the purpose. 



i,5(} 




Fig. 1063. 

Fig. 1063 shows Normandy's pipe joint. The packing con- 
sists of two rubber rings. This very simple joint is ver}' useful 
under certain circumstances, where the proper packing is avail- 
able. It possesses the flexibility of Petit's joint and is easily 
connected and disconnected. 

A similar form of joint has been made for water pipes, using 
packing rings of lead. The sleeve may be considered as a 
double bell and the pipes are perfectly straight without any 
bead at either end. The distance from the centre of one joint 
to that of the next constitutes a "length." With cast iron 
pipe this is made a minimum of about 4 to 7 feet, being made 
as long as practicable for extensive lines of pipe. For gas and 
water pipe with bell and spigot connections the following pro- 
portions occur in practice : 

/? =: 4 inches, / = 7 to 10 ft. 

" /= 10 to 12 ft. 

" and over, / = 12 ft. 

A form of joint used by Riedler for high 
pressure water connections is shown in 
Fig. 1064. t The flanges are faced in the 
lathe and bolted together without any 
packing in the joint. A leather ring is 
placed in a channel turned ir the pipe and 
held in place by a spring ring in two parts, 
or this latter ma)' sometimes be made in 
one piece. 

Joints with spherical contact surfaces 
have been used by Hoppe for cast iron 
high pressure pipes when they are to be 
laid in yielding ground, j 

Three forms of constraction are shown 
in Fig. 1065. At (Z is a single ball joint. 

* This joint is used by the Lauchhaminer Works for pipe up to lYz inches 
diameter. 

tSee Zeitschr. D. Ing, Vol. XXXH., i38o, p. 481. 
i German Patents, No. 42,126. 




Fig. 1064. 



The bearing ring is held in position by a ring of bronze divided 
at right angles to the axis ; this form permits a deflection of 5°. 
At b is shown a double joint constructed in a similar manner 




Fig. 1065. 

and permitting a deflection of 10°. The third form, which is 
the most recent, has no packing ring, and the bolts are made 
with spherical heads to facilitate motion. 

? 342. 

Cox>fECTioNS FOR Pipes of Wrought Iron and Steel. 

Riveted pipes are often connected by means of wrought or 
cast iron flanges, as shown in Fig. 1066 a and b. When no 

b 

i'l 




4a 



^M. 



■;r^ 



Fig. 1066. 



k-j^d--^ 



other data are at hand, the diameter and number of bolts ma}' 
be_ determined by assuming the pipe to be of cast iron, and 
using the proportions given in the illustration. The actual 
thickness i of the pipe may then be determined independently 
according to the material, pressure, and other conditions. 

Exajnfile.—A. wrought iron pipe 3 ft. 4 in. in diameter, for delivering water 
to a turbine, is to be fitted with flanges of wrought iron. A cast iron pipe 
of this diameter would have a thickness, according to (318) 

— « Q,-" 

80 

whence from Fig. 1056 c? = J X 0.815 = 1.05", say iji^. The number of bolts, 
according to {337). will be 2 -|- ^'^ = 22. If the internal pressure is 30 pounds 
per square in. we have, according to (324), taking .S= 4200; 

S = 0.5 ^° '''' *° = 0.143". saysV- 

4200 



LM 




Fig. 1067. 

For thin pipes a very practical form is that shown in Fig. 
1067 a. The ends of the pipes are flanged over, and the turned- 
over ends countersunk into the cast flange rings, the bolt heads 
also being countersunk. A similar form with wrought iron 
flange rings is shown at b. l For the thin pipes described in 
? 337' w-hen subjected to a high internal pressure, the joint 
shown in Fig. 1067 c is adapted. In this form a short sleeve is 
riveted into one of the pipe ends and a loose ring slipped over 
the outside of the joint, forming a space into which lead is run 
and afterwards caulked. This also serves as a sort of expan- 
sion joint (compare § 338). 

Many important constructions are made with wrought iron 
pipe. The connections are usualh- made bj' screwing the parts 
together, and for this purpose many special pieces are made, 
known by the generic term of " pipe fittings." For straight 
connections the ordinary "socket" is used, while for angles 
the so-called '"elbows " and " tees " are made. 



g For description of a flange joint with welded conical rings, by De Naeyer, 
sec Zeitschr. D. Ing., Vol. XXX, 1866, p. 106. 



250 



THE CONSTRUCTOR. 



The American practice of making the thread tapering is much 
to be recommended, since b}- means of a little cementing mate- 
rial a tight joint may be made. The American Mechanical En- 
gineers have given careful attention to the proportions of pipe 
fittings, and since 1887 the forms proposed by the late Mr. 
Robert Briggs have been generally adopted.* 

The system is as follows : The thread is of triangular section 
with the angle 2 ;3 = 60°, as in Sellers' system. The top and 
bottom of the thread are flattened Jj of the theoretical depth 
4, so that the actual depth t = 0.S4, and hence equal to 0.69 of 
the pitch J, Fig. 1068 a. 







v/Vj" 



Fig. 106S. 

The end of the pipe is given a taper of jV on each side, the 
length of the tapered part being 7"= (4.8 -j- o.?,D)s, D being 
the outside diameter of the pipe and .f the pitch. Beyond the 
taper portion comes a length 7", = 2 i~, which threads are full at 
the root but imperfect at the top, beyond which there is a 
length T^j = 4 i, consisting of imperfect threads blending into 
the full outside diameter. The thickness & of the pipe is such 
that the thickness of metal below the thread at the end of the 
pipe is = 0.0175/) + 0.025". The pitch .s is finer than for 
bolts of the same diameter, there being only five different 
pitches used, and the various dimensions are given in the 
following table : 



Tabi,b; op Standard Pipe Threads. 





Diameter of Pipe. 




Screwed Ends. 






Thickness 

of 

Metal. 






Nominal 
Inside. 


Actual 

Inside. 

D. 


Actual 

Outside. 

Do. 


Threads 
Per Inch. 


Length. 
T. 


Inches. 


Inches. 


Inches. 


Inch. 


No. 


Inch. 


% 


0.270 


0.405 


O.06S 


27 


0.19 


% 


0.364 


0.540 


0.088 


18 


0.29 


Vi 


0.494 


0.675 


0.091 


18 


0.30 


% 


0.623 


0.840 


0,109 


14 


0.39 


X 


0.824 


1.050 


O.I13 


14 


0.40 


I 


1.048 


I-315 


0.134 


11;^ 


0.51 


iJi 


I 380 


1.660 


0.140 


">^ 


0-54 


^% 


I.610 


1.900 


0.145 


II "X 


0-55 


2 


2.067 


2-375 


0-154 


II>^ 


0.58 


^y^ 


2.468 


2.875 


0.204 


8 


0.89 


3 


3.067 


3500 


0.217 


8 


0.95 


i% 


3.548 


4.000 


0.226 


8 


1. 00 


4 


4.026 


4.500 


0.237 


8 


1-05 


e/z 


4-508 


5.000 


0.246 


8 


1. 10 


5 


5-045 


5-563 


0.259 


8 


1. 16 


6 


6.065 


6.625 


0280 


8 


1.26 


7 


7023 


7.625 


0.301 


8 


1.36 


8 


8.082 


8.625 


0.322 


8 


r.46 


9 


9.000 


9.688 


0.344 


8 


1-57 


10 


10.019 


10.750 


0.366 


S 


1.68 



Taper of conical portion of tube i in 32 to axis of tube. 

It will be observed in the table that the thickness d agrees 
very well with the formula (5 = o.ii i ^D^. This gives for the 
diameters 0.405, 1.050, 4.000 and 10.750, the thicknesses 0.071, 
0.114, 0.222, 0.364, which agree quite closely with the actual 
values. The shape of the sockets is shown in Fig. 1069, the 
thread being given a somewhat greater taper than jV, so that 
the greatest stress will come on the strongest part of the 
socket. 

The increasing use of such pipe in Germany makes it most 
desirable that a standard of dimensions should be adopted. 
The American system is manifestly unsuited for use with the 
metric system. The general arrangement of the American 
system may, however, be followed with some approximations 
to adapt it to the metric measurements. 

The angle of thread may be the same as in the American 
system ; 2 /3 = 60°. The depth of thread may also be abbre- 
viated y'j top and bottom, making / = 0.8 i^^ = 0.68 .y, and the 



taper can also be made -}, on a side. The length T of the 
tapered portion may be niade 7" = (5 + ,ij j9„j j, which is 
about the metrical equivalent of the former expression, the 
nearest even value being taken. The lengths T^ — 2S and T-^ 
^ AS may be retained. 

For the thickness of pipe the American formula transformed 
gives S = 0.555 ^/7?o in millimetres. Finally for the pitch we 
may take 

s ^ I 1.4 1,8 2.2 3.2 mm. 

(0.94) (1.41) (I -81) (2.21) (3.17) in. 
the values in parentheses being the corresponding equivalents 
of the American pitches. The following table gives the values 
from 10 to 325 mm. This system has been submitted by the 
author to the manufacturers of the Mannesmann tubes in 
Remschied, Saarbriick and Komotau, and by them adopted. 

Metric Pipe Thread System. 



Outside 
Diameter 


Thickness 


Inside 


Pitch 


Length 
of Thread 


Length 


Length 


-Do- 


D. 




T. 


7i — 2 J. 


7; = 4 J. 


iO 


1-75 


6-5 


I.O 


5-5 


2.0 


4 


15 


2.00 


1 1.0 


1-4 


7.5 


2.8 


5-6 , 


20 


2.50 


15-0 


1.4 


8 


2.8 


5.6 


25 


2-75 


1 9-5 


1.8 


11 


3-6 


7-2 


30 


3-00 


24. 


1.8 


12 


3-6 


7-2 


35 


3-25 


28.5 


2.2 


14 


4-4 


8.8 


40 


3-50 


33-0 


2.2 


15 


4-4 


8.8 


50 


4.0Q 


42.0 


2.2 


■ 15 


4-4 


8.8 


60 


4-25 


51-5 


2.2 


16 


4-4 


8.8 


70 


475 


60.5 


3-2 


25 


6.4 


12.8 


80 


5.00 


70.0 


3-2 


26 


6-4 


12-8 


90 


5-25 


79-5 


3-2 


28 


6.4 


12.8 


100 


5-50 


89.0 


3.2 


29 


6.4 


12.8 


110 


5-75 


98.5 


3-2 


30 


6.4 


12.8 


120 


6.00 


108.0 


3-2 


31 


6.4 


12.8 


130 


6.25 


1 17-5 


3-2 


33 


6.4 


12.8 


140 


6.50 


127.0 


3.2 


34 


6-4 


I2.S 


150 


6-75 


136-5 


3-2 


36 


6.4 


12.8 


'75 


7-25 


160.5 


3-2 


38 


64 


1 2.8 


200 


7-75 


184.5 


3-2 


42 


6.4 


12.8 


225 


8.25 


208.5 


3-2 


45 


6.4 


12.8 


250 


S.75 


232.5 


3-2 


48 


6.4 


12.8 


275 


9-25 


256.5 


3-2 


51 


64 


I2.S 


300 


9-50 


281.0 


3-2 


54 


6.4 


12.8 


325 


10.00 


305. 


3-2 


58 


6.4 


12.8 



In the preceding table the pipe is classified according to its 
outside diameter /?o, but it is a question whether it would not 
be better to follow the custom of designating the sizes by the 
internal diameter D. The former, however, has an important 
influence upon the dimensions of the fittings, which it is most 
desirable to reduce to a standard system. It will be seen by 
reference to the table of American pipe dimensions that the 
actual internal diameter differs frequently from the nominal 
size, the latter really being only a convenient name. By adopt- 
ing a strict gradation for the sucessive sizes of D^ it would be 
practicable to make the thickness & somewhat less than given 
in the table, but in some cases it would be greater. When D^ 
is greater than 325 mm., & may in ordinary cases be made 
= 10 mm. 

The production of the screw threads both in pipe and fittings 
must be carefully considered in order to insure the interchange- 
ability which is necessary. Powerful and accurate machines 
have been devised for cutting the threads, as well as devices for 
producing the taps and dies, and also gauges to insure mainte- 
nance of standards. This branch of the art has been carried to 
a high degree of perfection in America. 

Fittings for Wrought Pipe. 
The simplest pipe fitting is the socket used for connecting 
two pipes of equal diameter D^, and is made of wrought iron 





* See Trans. Am. Soc. Mech. Engineers, Vol. VII, pp. 311 and 414 ; also Vol. 
VIII, pp. 29 and 347. 



FlG. 1069. 

or of steel. It is made of sufficient length to give a thread in 
each end of length equal to T, as given in the preceding tables, 
together with a slight clearance between the ends of the pipes, 
Fig. lodtja. In many cases the socket must be made with right 



THE CONSTRUCTOR. 



251 



and left hand threads, as in Fig. 106915, this being necessary to 
connect two pipes which cannot be turned axially. For other 
connections a variety of fittings are made, examples of which 
are shown in Fig. 1070. 




FlO. 1070. 

In Fig. 1070, a is a right angle ; b an elbow (abbreviated in 
practice to " ell ");<:• is a T ; rf a cross ; and e a reducing socket. 
These fittings are used as connections for all sorts of gaseous 
pressure organs. They may also be used for liquids, as water, 
brine, oil, etc., when the velocity of flow is not great. For im- 
portant installations it is becoming more and more the practice 
to design the fittings iu such forms as to produce a minimum of 
resistance. By making the fittings of cast iron, as is done in 
England and America, where pipe constructions are very ex- 
tensively used, it is possible to adhere to accurately designed 
standard forms. 

The most important fitting is the elbow, for the right angle 
bend occasions far too much resistance to be used in important 
cases. In Fig. 1071 three forms are shown, all of which are 

A i> c 



D i;>. 




D,;-T 



I r — 4-D--' i 



Fig. 1 07 1. 



designed to be used with the thread already described. Of 
these, form b is the most popular, although form a is frequently 
used because of the smoothness and neatness of external ap- 
pearance. Form c is here proposed as an additional design. A 
comparison between the three forms will show a difference in 
resistance which may be calculated as follows : The resistance, 
may be divided into two portions ; one due to the curvature, the 
radius of curvature being made equal to /?„ ,• and one due to 
the enlargement and consequent contraction of the passage. 

Example 2. — In the three forms shown in Fig. 1071 let the radius of curva- 
ture D^ = 1 inch, and let the velocity z> be taken at 6.56 feet per second. 
We then have from (335) for the resistance due to the curvature, h■^ = ^a 

— — . -^ — = 0.334 f'>, and for ^^ in the various forms 
90 64.4 



0.5 



D 



whence 



i2 



a 


h 


c 


0.66 


0.54 


0-39 


0.573 


0.352 


0.201 


0.191 


0.118 


0.067 



We also have from (336) for the loss due to enlargement and contraction : 
. 64.4 , 



/;, = : 



& 



]- 



336^3 



hence 




a 


* 




F 
Fx 


V 0.75 / 


/ 1.0625 
\ 0.75 




C, = 


3-444 


0.904 


hence 


/<3 = 


4.6 


1.207 



hence /in + ^'3 = 4 79^ 



1-325 



o 067 



It will thus be seen that form a cannot be recommended, except for steam 
for which the coefficient of loss is much less than for water ; and that form 
6 occasions quite a perceptible loss. Form c is much to be preferred, both 
because it offers the least resistance, and also because it is ligrhter, the pro- 
portion of metal in the curved portion of the three forms being as 36 : 30 ; 25. 

The only dimeusion which is important in connection with a 
standard system of fittings is the distance Z^q ~H ^' which 
should be taken from the preceding table. The thickness <5^ is 
mainly dependent upon matters of casting, and is here made 
= fj 4" 0.04^^ ((S -|- I mm.) the thickness of the collar being 

An indispensable condition for any standard system of fittings 
is the constant length from end to centre for each size of elbow, 
cross, or T, so that at any time one fitting may be substituted 



for another without affecting the length of the pipes. This 
principle can also be observed when the fittings are used to 
connect pipes of different diameters.f Such fittings are always 
known by the name of the largest opening, whether T, elbow, 
or cross, this dimension governing the proportions. 



'. I \: 





__Z1H±K <i : '" ■; : " 



Li.. 


-J D,.v 


'IG 


1072. 



-iHi 



Fig. 1072 a shows a T, which is proportioned to permit one- 
half the flow of water to pass off the side opening. This is 
based on the form b, of the preceding illustration, and, as is 
usual, the direct discharge opening is made the same size as the 
entrance. £>' is made equal to 0.7 L>, thus giving one-half the 
area, and making the velocity the same as in the entrance pipe ; 
if the side opening had been kept full the velocity would have 
been reduced one-half. The side outlet is shaped like an elbow, 
with a sharp internal partition to direct the flow. According to 
Roux, these partitions are of much importance, acting as wedges 
to split the flow of the water. At b is shown another form, in 
which both discharge openings are reduced, and every precau- 
tion taken to give a smooth flow to the water. At if is a reduc- 
ing fitting which will double the velocity of flow, the reduction 
in diameter being made by gradual curves. 



I . 1 , 

I ^ _ :■: i 3 

BJ.lL.irS^;^-?]]-.-!?" a 




TTJI j_ fjS3j .*.... 



-N 



— fDo-i 

Fig. 1073. 




Fig. 1073 a shows a T with equal outlets, formed on the plan 
of the elbow shown in Fig. 107 1 b. This is made with a divid- 
ing wedge, which is much better than the straight form shown 
by the dotted lines. The latter form causes material loss by the 
sudden reduction of velocity to one-half The form shown at b 
is intended still further to reduce this loss. At c is shown a 
cioss with three equal outlets designed on the same principle. 

The previously described fittings have been given on the as- 
sumption that the velocity of flow is to be kept uniform from 
the point of division both as regards the fittings and in the 
pipes. In extensive installations, whether iu residences, public 
buildings or manufacturing establishments, this is not often the 
case. Very often it is found that one portion of a system is 
possessed of but little velocity of discharge, while a neighbor- 
ing pipe has a flow of high velocitj' iu it. The resistances in 
such systems become quite material, but may be somewhat re- 
duced by giving care to the shape of the fittings. 

In adopting standard dimensions for pipe fittings, which may 
be based either upon form b or r, especial precautions must be 
taken to insure interchangeabilit}', this being the principal ad- 
vantage to be obtained. This involves accurate tappiug of the 
threads both in the sockets and in the right-angle fittings, 
which is accomplished by special devices which enable all these 
operations to be performed without releasing the fitting, the 
accuracy of angles and sizes then being readily controlled by 
the machine. The sizes of the fittings are cast upon them in 
distinct figures, so that they may readily be determined. 

? 343. 

CONNECTIOXS FOR PIPES OF LEAD .\ND OTHER METAI,S. 

Lead pipes may be connected by means of separate flanges 
of wrought iron which draw the expanded ends of the pipes 
together. 



* The small clearance for the screw thread may be neglected. 



t See Trans. Am. Soc. IVIech. Engrs., IV, p. 273. 



252 



THE CONSTRUCTOR. 



A good flange connection for lead pipe is shown in Fig. 
1074;* the pipes are expanded and a double cone socket of 
brass inserted and drawn together by bolts. Fig. 1075 shows 



J 


r 


1 




1 
1 


\w 


1 , 




■\ 


lllli 


il \^^~ 


-1 

11 ! 


^jfc 




i 


^ 


\ai 


■ ii 








|.57to.i.l) 



Fig. 1074. 



Fig. 1075. 



another design, by Louch ; the pipes are drawn together by 
means of screw flanges and a collar, the three pieces all being 
made of cast iron. 





Fig. 1076. 

Fig. 1076 a shows a connection for joining lead to cast iron 
pipe, and Fig. 107615 is for lead to wrought iron pipe ; the loose 
collars in both forms are made hexagonal or octagonal exter- 
nally, so as to be operated hy wrenches. 

I 344. 
Fi,ExiBi,E Pipes. 

For many purposes it is desirable to have a pipe which shall 
be yielding or flexible, so that, for example, it may follow the 
inequalities of the ground, or may accommodate itself to 3'ield- 
ing supports. In such cases the flange connections maj' be 
constructed to permit motion by means of ball and socket 
bearings, as shown in Fig. 1065, such joints being especially 
adapted for pipes to be laid under water. An example of such 
construction is found in the water main built by G. Schmidt, of 
Carouge, for the water supply of Geneva, laid on the bed of the 
Lake of Geneva. The pipe is 47 '^: inches (1.2 metre) diameter, 
and is made in lengths of 29^4 feet of riveted wrought iron, 
0.197 in. thick (5 mm.). The connections are ball and socket 
flanges, riveted to the pipes. 

Instead of making the pipe rigid and the joints flexible, the 
joints may be made rigid and the pipe flexible. Familiar ex- 
amples of flexible pipe are various kinds of hose, made of 
leather, canvas or rubber. Special forms of couplings are made 
for fire hose. If the hose is to be subjected to heavy pressure, 
either internally or externally, special methods of increasing 
its strength are used. This may be done by means of a spiral 
of wire, or better by two separate spirals, one to resist internal 
pressure and one to resist external pressure, as shown in Fig. 
1077 a. The wire spirals furnish the strength and the hose the 




Fig. 1077. 

tightness. This idea may be still further carried out by making 
the material which makes the pipe tight, also in the spiral form. 
This is shown in the flexible metallic tubing of Levasseur, of 



Paris, shown in Fig. 1077 3. f This is composed of a spiral of 
copper or similar metal, the section resembling somewhat the 
figure 5. The spiral is wound upon a mandrel in a special 
machine, a la3'er of rubber packing being wound in at the same 
time, as shown in the illustration. This pipe has been found to 
answer well for gas, water, steam, air, etc., and is adapted to 
high internal or external pressures, being tested to 180 pounds 
Flanges and other fittings are screwed on to the spiral and 
soldered carefully. This pipe is used, among other purposes, 
for connections for air and vacuum brakes. 

?345- 
Pistons. 

Next to the various kinds of pipes, as already discussed in 
? 310, the most important members in pressure organ mechan- 
ism are the various forms of pistons, and with these the differ- 
ent methods of packing will be considered. Pistons, properly 
so called, are fitted with packing which presses outward against 
the walls of the cylinder, while in the case of plungers the 
packing presses inward. Both forms will be given considera- 
tion. 

The most important forms of pistons are those used in steam 
engines. Some of the low-pressure engine pistons are yet 
made with hemp packing ; but for higher pressures, metallic 
packing is used, this consisting of metal rings pressed against 
the walls of the cylinder by springs and b}' the steam pressure. 
In some instances a combination packing is used, the metal 
rings having a backing of hemp instead of springs. 

The unit upon which the dimensions of the following pistons 
are based is determined from the formula : 



= 0-36S W /) — o 



04 — o. iiS 



(339) 



in which D is the piston diameter in inches. 

The following table will aid by giving a series of values for 
s and D : 



s 


n 


.s 


V 


.J 


£> 


0.4 


4 


0.65 


20 


0.90 


58 


0.45 


5-7 


0.7 


24 


0-95 


70 


°-5 


8 


0-75 


30 


1. 00 


S5 


0-55 


II 


o.S 


40 


1.05 


100 


0.6 


■ 14 


0.85 


48 


1. 10 


120 





0,9 


_fni 


K^ 


9^^3 




* 


^g.. 


<^^^: 

•.■:-;•-;■, 




Fig. 1078 shows a hemp packed piston by Pen?^. This is 
made of a cored casting with a ring follower secured by bolts, 
screwing into bronze nuts recessed into the piston. For pistons 
of large diameter an increased depth is given at the centre ; 
this increase may be made by making the depth in the middle 
equal to 6i + xV-''^. the depth at the edge being 7.8s, and the 
piston being made flat — when the latter value exceeds the 
former. 

Example. — Let Z> =^ 24 inches— for a hemp packed piston, as Fig. 107S, we 
then have s = 0.7. This gives for the thiclcness of the packing 0.7 X i-8 = 
1.26, say I'X in- ; the depth of packing = 0.7 X 6= 4.2 In. ; the depth of piston 
at tile edge = o 7 X 7.S = 5.36 = say 5-3 in. The depth in the middle will be 
equal to 6 X 0-7 X tS = 6.6, say 6;"s ins. 

Fig. 1079 shows a good form of piston with metallic packing, 
by Krauss. The packing consists of two steel rings, each cut 
at an angle, a ring of white metal being cast on each steel ring. 
If it is desired to make the cut in each ring tight, some one of 



* German Patent, No, 11,535. 



t Made by the Metallic Tubing Co., Ld. 
Loudon, N. C. 



Port Pool Eane. Gray's Inn Road. 



THE CONSTRUCTOR. 



2S3 



the methods shown in Fig. loSo may be used. In the iirst one 
the overlap makes a tight joint, while in the others the inserted 
piece is fitted steam tight. By filling the packing rings with 
white metal the wear comes mainly upon the softer material 
instead of on the cylinder, a most desirable feature, since the 
rings are easily and cheaply renewed. For the same reason 



A piston for a single acting engine, with combination pack- 
ing, is shown in Fig. 10S3. The metallic packing rings are 
backed with hemp, this combination presenting the advantage 




Fig. 1079. 

bronze rings are used, while iron or steel are not to be recom- 
mended, with the exception of soft cast iron, which works well, 
the cylinder being made quite hard. 

In Fig. loSi is shown the so-called "Swedish" piston, as 
used in a large blowing engine by Egestorff. This piston is 




&. 






r 


1 

1 


\^/7~ 


•-r;SrilJ 


< 


l« 


+ 16 D — *■ 



Fig. 1082. 

of elasticity together with durability. This style of packing is 
well suited also for marine engines, as its elasticity renders it 
less likely to be injured by the pitching and rolling of the vessel 
than an entire metallic packing. 




made with increased depth in the centre, similar to that in 
Fig. 107S, and the holes shown in the sectional plan view are 
for the purpose of removing the core from the casting. The 
packing rings are made of cast iron, with the joint made as 




Pistons for pump cylinders may be packed with leather so 
long as the temperature of the liquid to be pumped does not 
exceed 88° F. (30° C). 




Fig. ioSi. 



shown in Fig. loSo a. The rings are kept in their proper posi- 
tion b}' small pins. The method of securing the piston to the 
rod is worthy of notice. The large key is secured and tightened 
by a smaller ke}', the latter being held by a bolt, thus forming 
a fastening of the third order. 

Fig. 10S2 shows a metallic piston in which the packing rings 
are pressed out by an inner spring ring of steel.* 

The double cone shape of the inner ring enables the piston 
to be closely fitted to the cylinder by tightening the bolts when 
the engine is built. The nuts for the bolts are made of bronze, 
as in Penn's piston, the thread in this case being carried entirely 
through the nut and the hole closed by a plug. 



* E. Webers & Co., Machine Works, Rheiiie, Westphalia, 
makes a specialty of high class steam engines. 



This firm 



A form of packing for this purpose is shown in Fig. 1084, the 
principle being the same as the forms shown in the following 
section. The units for the dimensions are the same as already 
given. 

? 346. 
Plungers and Stuffing Boxes. 

As already observed, the packing for plungers and rods acts 
from the circumference inward, and such packings, in connec- 
tion with the necessary parts, are known as stuffing boxes, 

Two stuffing boxes for leather cup packing, especially adapted 
for hydraulic presses and for pumps, are shown in Figs. 1085 
and ic86, the former being for small and the latter for large 
plungers. The double cup in Fig. 10S5 is made with a spring 
ring of iron between the cups to hold them in position before 
the water pressure is applied. When the form shown in Fig. 
10S6 is used in the horizontal position, a ring of bronze made 
in several parts is introduced below the packing, as shown in 
dotted lines. This is intended to support the plunger and pre- 
vent it from rubbing against the cast iron cylinder. The propor- 
tions given in the illustrations are all based on the unit s, given 
by formula (339). 



254 



THE CONSTRUCTOR. 



The friction existing between a plunger or piston rod in the 
ordinary stuffing box in which the packing is tightened by 
screws, cannot well be calculated, as it depends upon the pres- 
sure which is put upon the packing. In those forms of stuffing 
box in which the pressure in the cylinder tightens the packing 
the friction may be calculated. According to the very elaborate 



s,6to 5,0 





Fig. 10S5. 



Fig. 10S6. 



researches of Hick,* the friction of a well-lubricated cup leather 
packing is independent of the depth of the packing, and is 
directly proportioned to the water pressure and to the diameter 
of the plunger. If P is the total pressure, D the diameter of 
plunger, and i^the fractional resistance, we have : 



~P 



0.04 



(340) 



For a new leather packing the friction is about ij^ times 
greater. If instead of the total pressure P we use the pressure 
p, in pounds per square inch we have : 

~P 



— = 0.0393 



D 



(341) 



Example. — For a piston rod 0.4 in. diameter, according to {340) the loss by 
friction would be ^^, or 10 per cent., while for a plunger 24 in. diameter it 
would be 0.0016, or y^ of i per cent. If, for example, the pressure is 4000 
pounds per square inch, the friction according to (341} would be 

F = 4000 X 0.0393 X 0.7854 X 24 = 2963 pounds. 
The total pressure on the plunger would be 

/*= 4000 X 0.7854 X 24- = 1,810,000 pounds. 

Stuffing boxes for the piston rods of steam engines must be 
capable of resisting the action of heat. Hemp packing is still 
much used for this purpose. The following illustrations show 
two excellent forms of stuffing boxes to be used with hempen 
packing. 




Fig. 1087. 



Fig. 10S8. 



Fig. 1087 is intended to be used on the top of a cylinder; 
Fig. 10S8 is for an inverted cylinder. Both gland and box are 
fitted with bronze rings, in order to reduce the wear upon the 
rod. The wedge-shaped edge which is given to these rings was 
introduced by Farcot, and is an improvement on the older style 
of beveling the edge in one direction only, the latter method 
often drawing the packing away from the sides of the box and 
permitting leakage. In some designs the edge is left square, as 
in Fig. 1090, or slightly rounded, as in Fig. 1089. 




Fig. ic 



Fig. 1090. 



Fig. 10S9 shows a form especially adapted to inverted cylin- 
ders. The construction will be apparent on examination, and 
it will be seen that the ordinary arrangement is reversed, and 
the glaud is cast upon the cylinder and the box containing the 
packing is made separate. This prevents water from the cylin- 
der from readily getting into the box. 

In order to prevent the gland from binding on the rod it is 
important that care should be taken to tighten both nuts 
equally. In large marine engines, for example, the nuts are 
made with worm wheels upon a common shaft. For small 
stuffing boxes this is accomplished by having the screw thread 
cut upon the outside of the box, as shown in Fig. 1090. This 
box is intended to be made entirely of bronze. The nut is 
made with six or eight notches in its circumference, to enable 
it to be turned by a spanner wrench. 

The dimensions of all the preceding figures are based upon 
the unit 5 given by the empirical formula (339). 

Exaviph. — For a rod 3 ins. diameter, according to (339) we get j = 0.36. 
The thickness of packing will then be 0.36 X 1.8 = 0.648, say Y^ in. The 
height of bo.\ for Fig. 1087 will be 0.36 X 12 ^ 4.32 ins., and for Fig. 1088 0.36 
X 21 = 7 56 ins., and so for the other dimensions. 

In horizontal stuffing boxes the length of the bronze collars 
should be made not less than 8 to 12 .j, in order to reduce the 
wear. The dimensions given in the illustrations may some- 
times be modified in order to conform to the thickness of ad- 
joining parts, so as to avoid difficulties in casting and shrinkage. 

In some instances the stuffing boxes for valve rods for steam 
engines are made in two parts, divided in a plane passing 
through the axis of the rod. The flange of the steam chest is 
then made in the same plane, so that with this construction the 
chest can be opened and valve and rod very conveniently re- 
moved and replaced. 

The large plungers for mine pumps are packed with hemp, 
the stuffing boxes having 4 to 8 bolts. 

More recently metallic packing has been introduced for 
stuffing boxes of steam engines. An excellent example is 







Fig. 1091. 



Fig. 1092. 



shown ill Fig. 1091, which is made by Howaldt Brothers, of 
Kiel.t The rings are made of white metal, in double cone 



* See Verhandl. des Vcreins F. Gewerbfleiss, i366, p. 159. 



t German Patent, No. 15,418. Over 9000 such boxes had been made up to 
888 : one of these had been running eight years without opening. 



THE CONSTRUCTOR. 



255 



pairs as shown, thus causing the pressure to be exerted alter- 
nately against the rod and the walls of the stuffing box. An 
elastic washer is placed between the gland and the first ring to 
equalize the pressure. Fig. 1092 shows the standard metallic 
packing introduced on the Prussian State Railways by Super- 
inteudent Neumann. This uses a single ring of white metal 
made in two parts. The pressure is obtained from a steel spiral 
spring placed in the bottom of the stuffing box, and acting 
against a bronze pressure ring. The whole is enclosed in a 
steel cylinder which, together with its contents, can be drawn 
out by inserting a hook into a T-shaped recess. The form 
shown in the illustration is intended for a valve rod, but a 
similar pattern is used for the piston rod. 

? 347- 

Pistons with Valves. 

Pistons with valves are used in lift pumps and in steam en- 
gine air pumps. An example of such a pistol,, with leather 
packing, intended for a mine pump, is shown in Fig. 1093. 




_i-_i 



Fig. 1093. 

The packing is composed of conical rings of leather and 
canvas, each three adjoining layers being sewed together. The 
pressure of the water acts to tighten the packing. The acid 
mine water often acts injuriously upon the leather packing of 
the pump pistons, and in such cases metallic packing, with 
rings of soft cast iron, is used. At F'ahlun, in Sweden, after 
many experiments the best material for packing was decided to 
be birch wood. The proportions for Fig. 1093 are based upon 
the unit i. A valved piston for steam engine air pump is shown 
in Fig. 984. 

I 348. 
Piston Rods. 

Piston rods for steam engines are usually made of wrought 
iron or steel, and recently compound rods of wrought iron sur- 
rounded with hard steel have been used. The rod i.: either sub- 
jected to tension only, as in single acting engines, or is alter- 
nately subjected to tension and compression, in which case the 
length and resistance to buckling must be taken into account. 
For short rods the same results are obtained for both condi- 
tions, but in no case should a rod subjected to alternate tension 
and compression be made lighter than a rod under tension 
only. 

a. Dimensions of Piston Rods, Tension only. 

D = diameter of cylinder in inches. 

p = pressure in pounds per square inch. 



The total pressure P on the piston will be Z'; 



■ p D''. In 



order that the stress on the rod should not exceed S500 pounds 
we have for the diameter d of the piston rod when made of 
wrought iron, and is subjected to tension only ; 



d^ 
~D 



= o.oioS 



\f / 



(342) • 



(343) 



or for a close approximation : 

d^ ^ 57^±_°l5^ 
D 1000 

Example. — If / = 60 pounds we have from {342), — — - = 00836, and hence 

for a 20 inch cylinder c? = 20 X 0.0836 = 1.67 in. The approximate formula 

(343) gives — — T-^ = 0.0S7, which for /) = 20 ffives d = 1.74 in. 
1000 



Steel rods subjected to tension only may be made 0.8 times 
the diameter of wrought iron rods. 

If a piston rod is weakened by having a keyway cut through 
it, or by a screw thread, the reduction in cross section should 
be provided for by a proper increase in diameter. For this 
reason the diameter of the rod is sometimes increased in the 
cross head, an example of which will be seen in the locomotive 
cross head. Fig. 539. This construction involves the necessity 
of making the stuffing box glaud in halves, as it could not be 
slipped over the enlarged end of the rod. 

b. Dimensions of Piston Rods for Buckling Stresses. 

Using the preceding nomenclature and indicating the length 
of stroke by L, we have : 



d_ 
D 



: 0.0295 



(iH' 



(344) 



from which the following table has been calculated : 



L 


/= 50 


= 60 


= 70 


= 80 


= 90 


= 100 


= 120 


= 140 


= 160 


= 180 


1-5 


0.0967 


0,100 


0.104 


0.108 


O.III 


O.I 14 


O.I 19 


0.124 


0.129 


0.133 


z.o 


O.III 


0.II6 


O.I2I 


0.125 


0.128- 


0.132 


0.I3S 


0.143 


0.J48 


0.153 


2.5 


0.124 


0.130 


0.135 


0.140 


0.144 


0.148 


0.154 


0.I6I 


O.J66 


0.J71 



These values will serve both for wrought iron and for steel 
(compare I 1S2, and table in 'i 2). 

Example. — For a steam cylinder 16 in. bore, 4 in. stroke, with a pressure 
of 60 pounds, we have —^ = 2.5, and d = 0.130 X 16 = 2.08, say 2 inches dia- 
meter, either for steel or wrought iron. 

The dimensions of steel keys to secure the piston to the rod 
are so taken as to give shearing stresses from 5600 to 7500 
pounds in the key. Care should be taken that the key be not 
made too narrow, and the consequent superficial pressure be- 
come too great. Pressures of 6000 to 7000 pounds per square 
inch are found in stationary engines, and 10,000 to 15,00(7 
pounds in locomotive engines. 

'i 349- 
Specific Capacity of Pressure Transmission Systems. 

Having discussed the subject of conductors for pressure 
organs, we return to the consideration of the various mechani- 
cal devices which maj'be operated by pressure organs, although 
these have already been described in Chapter XXIII. We are 
now prepared to consider these in connection with the subject 
of long-distance transmission of power, in a manner similar to 
that in which tension organs are used in Chapter XXI. For 
this purpose we may use to advantage the conception of specif c 
capacity. This method is especially desirable because its sim- 
plicity and general character enables comparison to be made 
between widely differing systems. 

The conception of specific capacity can be extended without 
difiiculty to motors operated by water, air, steam, etc., since for 
all these we may put the general equation : 

A^o=^ 
qv 

deduced in § 280. In this equation g represents the cross sec- 
tion of the pipe or other conductor in square inches ; the mean 
velocity in feet per minute = i\ and yV being the horse power. 
If, for example, in a water pressure engine, h is the available 
head of water. Q the weight of water delivered per minute, and 
//' the head equivalent to the resistance against which the water 
leaves the engine, we have for the work delivered : 

33000 

p,ut Q = 0.0361 X 12 qv^= 0.434 '/ ^', the coefficient 0.0361 being 
the weight of a cubic inch of water, and the pressure p with 
which the water acts ^ 0.434 //, whence // -= 2.^p. Substituting 
these values we get : 

0.434^1' X 2.3 (/—/O I 

— qv{p—p') 



N=- 



33000 
and the specific capacity becomes 



33000 ■ 



ip-p') 



(345) 



a value of the same form as that previously deduced in ? 280 
[see formula (262)]. 



■56 



THE CONSTRUCTOR. 



Example. — If the effective pressure/ — /' be 320 pounds, the specific ca- 
pacity will be .'Vo = 0.0097. If the pipe is 4.75 in. diameter, and the water has 
a velocity of 236 feet per minute, we have ; 

N = 4-75 - X X 236 X 0.0397 = 40.56 H. P. 

This is only the capacity of the pipe. The effective capacity will 
be considered later. 

Formula (345) can also be used for air pressure or for vacuum, 
for steam or gas, by expressing the effective pressure in terms 
of an equivalent head of water. For steam and air it may be 
considered as an expression of the following form : 



vV„ = - 



(/■-/O^ 



(346) 



The coefficient /( is very comprehensive ; it increases with p 
and with the rate of expansion f, and can be calculated from 
these data, and also confirmed by observation. For f = 2, it 
ranges from Ij4 to 1-3, and increases to 3 to 4 for e = 20 to 30, 
results which conform to the higher pressures and greater effi- 
ciency of compound engines in which such high expansion 
ratios are used. 

With some transformations the equation for specific capacity 
may also be used to solve another important problem, that is 
the question of the best material to be used for the conducting 
pipe. 

If we assume the diameter of the pipe, the horse-power N 
will be : 

4 33°oo 

For the thickness of pipe, we have from 
the material : 



21), for a stress S, iu 



2&-Y D = D\ --.-^ 



V 5-/ 



And since 2 (5 -|- /? is the external diameter D^^ 
the cross section q^ of the pipe. 



we have for 



?i 







-((^o 



. z?M = 



U' 



(I^^O 



4 -3 —p 



Substituting the value of — £>- from its equation in the above 
4 



expression for jV, we have 



I 

000 



S — p 

1^-rp-p^' 



33000 V J 



'/i "■J 



whence 



iV„ 



Qy"" ~~ 33000 



(-0 



(347) 



a form similar to the preceding expressions for A^^- 

This expression is very instructive. It is applicable to all 
forms of conducting pipes for power transmission. It shows 
clearly the importance and value of a high value of .S. A high 
value of 6" reduces the proportional influence of /, to a degree 
which practically makes N„ dependent mainly upon ,5. It fol- 
lows that we may consider that the specific capacity of the pipe 
in a pipe transmission system, is practically independent of the 
pressure of the fluid used iu it. In other words, the capacity of 
a given pipe in horse-power is the same, whether the medium 
be liquid or gaseous, high or low pressure, provided the stress 
iu the material of the cross section of the pipe is constant. 

It is therefore desirable to use pipes of small diameter and 
fluids at moderately high pressures. The friction in the pipe 
need not prevent this, as care iu avoiding sharp bends and 
angles can be taken ; and as already shown in § 340 the friction 
is independeut of the pressure of the medium, at least so it 
appears from such experiments as have yet been made. 

The value of the stress in the material of the pipe cannot be 
taken very high ; S= 7000 lbs. being about the upper limit, 
and 5* =: 6500 lbs. appears to be quite high enough. Wrought 



iron and steel, especially in the Mannesmann rolled tubes, per- 
mit the use of high stresses; for wrought iron .S= 17,000 lbs. 
and for steel 35,000 to 40,000 lbs., or even higher, if necessary, 
may be used. By neglecting the value of/ in formula (,347) we 
have for : 



Cast Iron, 
Wrought Iron 
Steel 



5= 6,500, N^ 
S = 17,000, iVo 
•S" = 35,000, iVo 



: 0.197 
: 0.515 
: 1.060 



This gives an indication of the efficiency of the pipe system of 
power transmission and enables comparisons to be made with 
other systems. 

{ 350. 

The Ring System of Power Distribution with Pipe 
Conductors. 

Before proceeding with the further discussion of the preced- 
ing equatious it is advisable to investigate further the subject of 
power transmission by means of pipe conductors, as already in- 
dicated in ^ 312. It was there remarked that pressure organs 
might be used in connection with pipe conductors so as to form 
"ring" transmission sj'stems in a manner similar to those 
already described for rope. Taking into consideration first, 
hydraulic systems, especially high pressure hydraulic systems, 
we find two distinct kinds of "ring" systems which may be 
used. 








O 

J! 



Fig. 1094. 

In the first method, Fig. 1094, the flow of water under pres- 
sure starts from the power station T^, with a pressure p^, and 
proceeds to the first station 7",, where it operates a water pres- 
sure engine, and passes on with a reduced pressure pi. It has 
therefore operated at the station 7", with a pressure p^ — p^. 
With the pressure/; it passes on to the .■■econd, third, fourth — 
— nth station Tn, each time losing pressure until it returns to 
the power station with a final pressure p «, where it is again 
raised to the initial pressure of p^. This is practically a coun- 
ter part of the rope transmission system of Fig. 917. It is 
apparent that the water pressure engines (escapements) at Ti, 
7";, T^, — ■ — ■ — Tn, should all be of equal size in order to uti- 
lize the entire flow without excessive resistance. Automatic 
regulation, such as Helfeuberger's, described in § 328, is also 
desirable.* 




E P 



Pi A 



Fig. 1095. 

The second system is shown in diagram in Fig. 1095. It will 
be seen that at each station there is a branch or shunt tube, 
leading through the motor (or escapement) T„, and then re- 
uniting with the main pipe. The main pipe A, forks at the 
station iuto the two branches .B and C, of which the first diverts 
any required fraction of the power of the main flow, as j^j, ^, J, 
as the case may be. At the fork is a swing valve C', operated 
by a speed governor /?, driven by the motor. This governor 
requires the assistance of some form of power reinforcement, 
such, for example, as shown in Fig. 1037. The discharge pipe 
£> of the motor unites with the by-pass C, to form again the 
main couductor £. At the entrance in the main pipe A, we 
have the pressure />[ of the original flow ; the motor 'J\ is now 
supposed to be stationary, the stop valve at i?' having been 
closed by hand. The flap valve O which has been disconnected 



* The London Hydraulic Power Company has installed separate ring sys- 
tems, each with a single generator and motor. 



THE CONSTRUCTOR. 



>-':>7 



from the regulator before stopping the motor, is also closed. 
The flow of water then passes through C to i? with the pres- 
sure /,. 

When the motor T.^ is to be started, the valve B' is opened 
and the flap valve C gradually opened until the motor begins 
to move, when it is connected to the governor, which regulates 
it thereafter so as to keep the motor at its normal speed. When 
a heavy load is thrown on, the valve is opened so that the pres- 
sure p., in B, becomes a greater fraction of /„ and when the 
work is less it is reduced. The pressure of discharge/, acts as 
a back pressure so that the motor works with an effective pres- 
sure p2 — Pi- The flow of water in the by-pass pipe C, also 
passes the valve C with a pressure p.^, and unites with the dis- 
charge at E to be further utilized at subsequent stations until 
it returns to the power station, where if it has reached the min- 
imum pressure, it is permitted to flow into a tank, from which 
it is again drawn by the pressure pumps. If the return water is 
delivered under pressure it may be allowed to enter the suction 
pipe of the pressure pumps direct and so form a closed ring 
system to start anew on the circuit. 

This system has not yet to the Author's knowledge been put 
into practical operation.* 

The ring system of h3'draulic power transmission is to be 
recommended when the various stations are distributed over a 
wide area and are readily connected by a continuous line of 
pipe. The pipe can be kept from freezing in winter by occa 
sional gas flames, as has alreadj' been demonstrated by exper- 
ience with Armstrong's hydraulic cranes. The ring system 
should be carefully distinguished from those forms in which 
the flow of water passes through the motor and is allowed to 
flow off at lowest pressure of discharge. A corresponding dis- 
tinction is to be made with other forms of power transmission. 

The author distinguishes as "line" transmissions, those forms 
in which the transmitting medium does not return to itself in a 
complete circuit, in contradistinction with the " ring " systems. 
The older form of rope transmission (J 297) is therefore a " line" 
system, while the system devised by the author and discussed 
in ^ 301 is a "ring" system. A hydraulic system in which 
there is a free discharge of water from the motors is in like 
manner a hydraulic "line'' transmission system. 

There is, however, an intermediate form possible, namely, 
that in which water after passiug through a series of motors as 
in a ring system, is discharged freely from the last motor Tn. 
A similar arrangement is possible with other systems of trans- 
mission. We may therefore extend the definition of a "ring " 
system to include those forms in which the medium of trans- 
mission returns to the place of starting. The distinction can 
then be made between "open" and "closed" ring systems, 
the latter being shown in diagram in Fig. 917. 

High pressure hydraulic systems are well adapted for large 
railway stations where numerous elevators as well as winding 
hoists and other rotative machines are to be operated. For 
such installations a combination of "ring" and "line" systems 
is best suited. The hydraulic elevators are more conveniently 
arranged on a line system than in a ring circuit. An apparent 
objection to the use of high pressure water to direct acting ele- 
vators lies in the fact that the diameter of the plunger becomes 
so small as to be hardly stiff enough to support the load on the 
platform without buckling. This difficulty is readily overcome 
by use of the hydraulic lever, as shown in Fig. 956 a, the con- 
struction of which offers no difficulties, and it is unnecessary to 
go into details. 

Up to the present time air has only been used upon line sys- 
tems, either with direct pressure or with vaacum. Gas engines 
can only be operated on a line system since the gas is burned 
in the engine. Steam has been used in a ring system in New 
York for some time, on a long distance transmission, and short 
ring systems exist in most cases of compound or triple expan- 
sion steam engines as used in marine and stationary practice. 



T- ^ 

D=ra 




T, 










K, 


K "■ ~l 




Fig. 1096. 

Steam at a high initial pressure is expanded successively in one 
cylinder after anether, and between the last cylinder or station 
Tn and the first, or boiler T^, is placed the surface condenser 



7,n, where the medium reaches the minimum pressure and is 
converted into water to be returned to the boiler and start anew 
on the circuit. In order that the velocity of flow shall be uni- 
form the successive passages for the expanding steam should 
be made with continually increasing cross section as shown in 
diagram in Fig. 1096. If a jet condenser is used instead of a 
surface condenser the circuit becomes an open ring. The high 
economy which has been attained by the application of the 
" ring " system with steam in the form of multiple expansion 
engines, points to the possibility of a similar economy in the 
application of the ring system to wire rope transmission. 

Lehmann's hot air engine, which is a true closed circuit, is 
an example of the ring system confined within the limits of a 
single machine. 

§351. 

Specific Capacity op Transmission by Shafting. 

The subject of the specific capacity of shafting was not con" 
sidered in Chapter IX, and it is introduced in this place in 
order to obtain a basis for comparison with the other systems 
of transmission. 

If we have the moment PR and shaft diameter d, we have, if 
S is the fibre stress at the circumference 

16 

(see ? 144)- 
If we make the lever arm i? := J d, we have P = the force at 

the circumference of the shaft and hence P^=i S -^ d^. Taking 

V = the velocity at the circumference of the shaft and iV the 
number of horse-power transmitted, we have : 



N= 



Pv_ 
33,000 



^S—d^'-v 

2 4 

33.000 



But — d'^ = g, the cross sectional area of the shaft, whence 
4 



2 33000 



(34S> 



and hence the specific capacity of the shaft is : 



qv 



2 33,000 ' 



(349)- 



This expression, which is of the same form as those already ob- 
tained, does not give values numerically great, because S must- 
be taken low enough to avoid excessive torsion of the shaft. If 
we require, as in I 144, that the torsion shall not exceed 0.075° 
per foot of length we must have S < 630 d which gives for 
shafting from 2 to 6 inches diameter .S = about 1200 to 370c 
pounds and the specific capacity 

A^o= o.oiS to 0.056 (350)- 

In other words, such a shaft will transmit, at one foot per min- 
ute circumferential velocity, o.oiS to 0.056 horse-power for each 
square inch cross section. 

In the application of shafting to long distance transmission 
the friction of the journal bearings is a very important consid- 
eration. The influence of friction may be determined in the 
form of a general expression in a similar manner to that of the 
friction of water in a pipe {§ 340). According to formula (100) 
we have for the force P", exerted at the circumference to over- 

4 
come the journal friction i^^ — y times the weight of the 



shaft, that is = ^^f 



tO 



12 /, X 0.2S in which L is the 



length of the shaft in feet, and 0.2S is the weight of a cubic 
inch of wrought iron. It follows that the horse power iVj re- 
quired to overcome the friction will be : 



Fv 



X ? X 12 Z X 0.2S 



♦See the Author's article in Glaser's Annalen, Vol. XVII (1885), part 12. 
from his paper to the Verein fur Eiiseubahnkunde, Nov. 10, 1885. 



33000 33000 

and if we take the coefficient of frictiony"^ o.oS we have 



i'SS 



THE CONSTRUCTOR. 



' linrtn tt 96,422 



j3000 - 

N, = --■ 
96,422 

and if we wish the specific frictional resistance, we have : 



■qv 



' ^Jo qv 96,422 



(351) 



(352) 



This resistance is by no means inconsiderable. Expressed as a 
percentage it will be : 



so that in both instances it is less than one-fourth the resistance 
of the corresponding solid shafts, as given in (354). Hollow shaft- 
ing thus greatly extends the capacity of shafting for long dis- 
tance transmission and also permits an important economy in 
material. 

The subject of shafting made of steel tubing was not consid- 
ered in Chapter IX, and a brief discussion will therefore be 
given here. 

Let d^ be the outside diameter, d^ the inside diameter, let the 

ratio '- ^ V- Making i/" = 0.9 as is usual in practice with such 

tubing, the diameter for resistance to torsion, (compare formula 
1,133) ) will be: 



pr = - 



N, 



qv L I 

N ~ "96,442" ■ ^o 



353) 



'A'' ~ 96,42 

The value pr, it will be seen, is inversely proportional to the 

specific capacity. If we apply this to (.350) we have for a 2 inch assumed. If instead of 71, the circumferential velocity v, be 

shaft "■■■" "■^— "'■"-" '-'*■ 



P' 
and for a 6 inch shaft 



L 


L 


o.oiS X 96,422 


1735 


L 


_ L 

■" 5400 


0.056 X 96,422 



^0 = 0.39^^^ = 6.18,^^ (359) 

This requires that the number of revolutions be known or 
assumed. If instead of 71, the ci 
given, we have for the same shaft : 

do = 7.25 ^^ (360) 



(354) 



£', being expressed in feet per minute at the circumference of 
the shaft. The number of revolutions will be : 



3.S2 



hence 1735 feet and 5400 feet are the limits of length respec- d^ 

tively for the two diameters given, at which the frictional resis- 
tance will equal the total transmitting capacity. Much higher The diameterTor strength (compare (131)) will be 
efficiency is obtained by using hollow steel shafting such as is 
now produced by the Mannesmann process of rolling weldless 
tubing. This furnishes a seamless tube, of sufficient truth as to 
cylindrical shape, the journals of which may be made either 
entirely of steel or of so-called "compound steel."" 



= 1.11^^^= 5.35 <f 



.S' 






(361) 



(362) 



If we take the ratio of outer to inner diameter V ^ -r = 0.9 

do 
(compare ? 90) and the thickness of the journal d' = 0.4 d^ we 
have for N„ : 



If V is not assumed as above, it may be taken at will and the 
following formula used : 



nS 



No 



N 



T-i^oO + *0 ^'''^ 



in which, when 
a 



-which for V = 0.9 gives 

No = 1. 81 X 



phen : 
-^- = 1/' = 0.4 0.5 0.6 0.7 0.750.800.85 0.90 



• (363) 



_5_ 
33000 ' 



(356) 



^i — f 



1. 01 1.02 1.05 1. 10 1. 14 1. 19 1.242 1.427 



"which is decidedly higher than for the solid shaft. (The value 

S <^ 630 do must be retained to avoid too great torsion). For The weights of tubular and solid shafting are to each other as 

the frictional resistance at the circumference of the shaft we ^j 'X /" n 



iave : 



F= 



.and if d' =^ 0.4 do we have : 

0.4 Z ^ : 



f- ]12 L X 0-28^- 



] 



Example. — If iV= 60 horse-power, k = 120 revolutions per minute, we 
have from (359) 



. 4/60 
> 120 ^ 



instead of 



jv, = . 



96,422 



241,000 



- q V 



' 60 



or dividing again by N : 



/..= 



L 



N 241,000 N^ 



(357) 



With the values for N^ as given in the two preceding instances, 
we have for the 2 inch shaft : 



do = 6.184 

<* = ■t-7 \f j;^ = 3-95 in. 

as would be the case for^ a solid ^shaft. The hollow shaft, however, weighs 

only 

\^'^^ J \ J 

120 X 5= 



I — — III — o-8i I = 0.33 times the weight of the solid shaft. 



The 



circumferential velocity -v = - 



3.82 



- = 163 feet. If a higher speed be 



N^ = 0.0326 




and for the 6 inch shaft : 




N^ =-o.ioi 




and these give in (357) : 




for the 2 inch shaft — 


■ 


0.0326 X 241,000 


L 

- 7S56 


and for the 6 inch shaft — 




*.- ^ 


L 


o.ioi X 241,000 


24,341 



chosen, as may readily be done, on account of the small journal diameter 
d' , we have from (360), making v = 300 ft., for example: 



do = 7 25 



sf, 



whence d' = 0.4 d^ = i-7 in- 

The number of revolutions will then be 



(358) 



3.82 X 300 

n = ^ — = 270 

424 



The weight ollshaft^will be 



(ig)-(— ) = •■■ 



* The Mannesmann " compound " steel tubing- is made with the interior times that of a solid shaft at 120 revolutions. The loss from friction will be 
of soft wrought iron and the outside of hardened steel. onlj- 0.26 times that of the solid shaft. 



THE CONSTRUCTOR. 



259 



Spfcific Vai<ue of Long Distance Transmissions. 

In the two preceding sections, equations have been given 
showing comparative relations between various methods of 
transmissions but at the i?»me time the general equation by 
which all the various methods of long distance transmission 
may at once be compared, has not yet been given. The point 
■which yet remains to be determined is the amount of material 
■which the tranimitted force carries in the shape of the trans- 
mitting medium. Investigation reveals certain fundamental 
points which may be applied either to a special case or to a 
comparative judgment as to the value of different sj'stems. 

The amount 0/ material required for ike principal transmit- 
ting medium of a long distance transmission system, may be 
considered as a function of the number of horse-power required 
to transport one pound of the material of which the conductor 
is composed over the distance between the origin of power and 
the point of application. 

The name ' Specific Long Distance Value " may properly be 
given to this quantity. If it is high, the method is efficient, if 
low it is less efficient for applications in which the distance 
plays an important part. 

In all the cases considered the medium of transmission may 
be taken as a form of prism of constant cross section q, having 
an endlong motion and the length of which is equal to the dis- 
tance A from the point of origin to the point of application. A 
chosen length A^ may be selected as a unit. The weight G of 
such unit will then be : 

G = 12 A^q a (364) 

in which a is the weight of a cubic inch of the material of which 
the conductor is made. The work in horse power exerted dur- 
ing the passage over the distance A^, is expressed bj- : 

N ^ N^q V 
Dividing this value by the preceding we obtained the desired 
result. It is desirable to select a standard unit for A^ which 
shall be generally applicable. Making A^ ^= i inch = -}^ ft, 

N 
and letting the quotient — - be represented by N^, we have for 

the general equation of "specific long distance value : " 



and also : 






■ . . .(365) 

These values hold good for all the systems which have been 
considered with the exception of rope and belt transmissions. 
In these the tension organ moves not only forward, but must 
be returned back on the slack side, and hence for these cases 
■we must put : 



A,=^ ± 



(366) 



When rope is used on the "ring" transmission system, how- 
ever, the preceding formula (365) is properly used, since a 
single rope makes the entire circuit back to the origin of power 
and the last point of application usually lies near the starting 
point. 

The formula for jVi is especially noteworthy because its 
application reveals great and unexpected differences between 
the various systems of long distance transmission. 

When the conducting medium is operated at a high velocity 
and also at a high working stress the specific value is very 
high ; when they are both small the specific value becomes 
lower, since they are both multiplied together. The following 
table shows both numerically and graphically the "specific 
long distance value " for the several methods named : 



The stresses in the table are laten at the maximum values 
given in the preceding pages of this work, without approaching 
too close to the upper limit. The high value of steel cable trans- 
mission is most noteworthy, and explains its frequent use. 
The table does not even give as good a standing for wire rope 
as might have been done, as when used with a tightening pul- 
ley its specific capacity is increased il times, see formula (310). 

Equally noteworthy with the value of wire rope is the poor 
showing made b}' shafting, especially solid shafting ; as it occu- 
pies the lowest position of all. The newer system of hollow 
steel shafting stands somevifhat better, but still very low. 

Welded iron and steel tubing when used for conductors give 
fairly good results. To avoid misunderstanding, it must be 
noted that where pipes are referred to in the table, v is the 
velocity of the fluid passing through them. No distinction is 
made between pipes for steam; compressed air or water, since it 
is only the weight of the tube which is here considered. A cir- 
cumstance worthy of note is that the reciprocal of A^s is pro- 
portional to the weight of the transmitting material, omitting 

N' 
connections, flanges, couplings, etc., so that Gs = -r;-. If, for 

As 

example, 200 horse-power is to be transmitted by hydraulic 
pressure over a distance A, of 9S4 feet = ii,So8 inches; the 



weight of the bare steel pipe will be 



1 1, 80S + 200 

144S 



= 1631 



pounds, the thickness of the pipe being so made that the stress 
on the material shall be 34,000 pounds, and the diameter such 
that the velocity shall be 7S7 feet per minute. Such calcula- 
tions are very useful for a general and preliminary investiga- 
tion. The designer must be careful not to lose sight of the fact 
that the stress in the material, and the velocity bear a very im- 
portant relation to each other. 

The values of jVs, given above, are the gross values including 
the entire work transmitted by the system. The net value 
(A'j)! and its relation to the gross value, that is, the quotient 

A^, . 

-pr is the next question to be answered. This question is by no 

o- 

means so simple as the preceding. The actual efficiency of a 
long distance transmission depends so much upon the resis- 
tances of friction, stiffness, centrifugal force, heat, etc., all of 
which differ for the different constructions, that only a very 
general allowance can be made to include them. A brief glance 
can only here be given to the method of determining this 
point. 

The greater the number of horse-power which can be trans- 
mitted for each pound of material, the less, proportionally will 
be the load upon the bearings and other points of loss, and 
hence the smaller, proportionally, will be the loss of friction and 
other hurtful resistances. In other words : 77/,? greater the 
specific value of the system, the less, in general, will be the pro- 
portion of hurtful resistance. 

The values already given in the table for the gross specific 
value, give also, therefore a measure of the net efficiency as 
well. 

While it can hardly be asserted that the above values for A's 
are inversely proportional to the losses from hurtful resistan- 
ces, yet there is a relation existing between them, so that it 
may be said that the net value (iV,)^ is in all cases higher than 
the gross value N s ; higher in the sense, that, the greater gross 
values are accompanied also with a higher net efficiency. 

The difference will appear most distinctly by comparing wire 
cable transmission with solid shafting. Such a comparison is 
the more readily made because in both instances the resis- 
tances can be closely determined. 



SPECIFIC VALUE FOR LONG DISTANCE TRANSMISSIONS. 



SYSTEM. 



Steel Cable, Ring System . . 
Steel Cable, Line System . . 
Iron Cable, Ring System . . 
Steel Conducting Pipe . . . 
Iron Cable, Line System . . 
Leather Belting, Line System 

Wrought Iron Pipe 

Hemp Rope, Line System . . 

Cast Iron Pipe 

Hollow Steel Shafting . . . 
Solid Shafting, Iron or Steel 



V 


5 


ff 


^'0 


Ns 


1 59°o 


21,000 


0.32 


0.318 


,S6S3 


i 5900 


21,000 


0.32 


0.318 


2631 


I 5900 


8,500 


0.32 


0.129 


23S0 


7S7 


34,000 


0.2S 


0.515 


144S 


5900 


8,500 


0.32 


0.129 


1 190 


5900 


540 


0.036 


0.006S 


1114 


7S7 


17,000 


0.28 


0.257 


722 


5900 


240 


0.036 


00036 


295 


7S7 


6,400 


0.28 


0.097 


272 


394 


5,200 


0.2S 


0.114 


160 


197 


4,200 


0.28 


0.C63 


45 



PERCENTAGE OF EFFICIENCY. 




PER 
CENT. 



100. 

50- 
40.6 
24.7 
20.3 
19- 
12.3 
5.02 
4.64 
2.73 
0.8 



26o 



THE CONSTRUCTOR. 



The iron wire cable transmissiou at Oberursel, discussed iu 
\ 300, showed a loss of about 14 per cent, in a transmissiou of 
104 H. P., over a distance of 316S feet. To transmit the same 
power over this distauce with solid shafting, we get from (353) 
the frictioual resistance : 



Pr = - 



J|6S_ T_ 

95,422 ■ vV„ 

Taking jVq = 0.063, "which is amply high enough, we have/ir = 
0.52 or about h. The net specific value for long distance will 
then be for iron cable, (i — 0.14) H90 =: 1023 ; for solid shaft-- 
iug, (I —0.5) 45 = 22.5. 

It will be seen that about 52 H. P. is absorbed in the friction 
of the shaft ; so that at periods of low water, when the turbine 
yields only 40.3 H. P. it would not be able to overcome the 
friction of the shaft alone. 

The correctness of these considerations will be confirmed 
when it is remembered that a wire cable runs at a very high 
velocity and operates at a high stress which the journals of the 
rope pulleys move at a very low velocity (scarcely t^j v) ; while 
on the other hand the shaft can only be subjected to a low 
stress, and the velocity at its circumference is not only low, but 
it has to overcome the resistance of friction at the same veloc- 
ity. This also explains clearly the reason why rope transmis- 
sion has so frequently superseded shafting iu actual practice. 



CHAPTER XXV. 
RESERVOIRS FOR PRESSURE ORGANS. 

I 353- 
Various Kinds op Reservoirs. 

Reservoirs form a most important feature in connection with 
the use of pressure organs, and are divided into tanks, receivers, 
chambers of various kinds, in which the pressure organs may 
be stored iu greater or less quantity and drawn upon for use as 
may lie required. Such reservoirs may be used either for posi- 
tive or negative pressure according to the system with which 
they are used. Both kinds are shown in Fig. 993, in the case 
of a canal lock. As already indicated in ? 312, the various 
forms of reservoirs are very numerous. From the nature of the 
subject we can only here discuss that branch of the subject 
which relates to machine construction, including reservoirs of 
cast and wrought iron, copper and steel. These are applicable 
both to gaseous and liquid organs and in most cases are of 
sp-^cial construction to meet the circumstances of use. 

A reservoir when considered in connection with the appar- 
atus for fdling and emptying, as well as for controlling the 
pressure, whether positive or negative, forms a storage system 
which mav properly be considered as a ratchet traiu (see Chap. 
XVIII). 

For the present, however, it is here only the intention to 
discuss the constructive features of the reservoir itself consid- 
ered as a machine element. 

2 354- 

Cast Iron Tanks. 

Cast iron tanks with flat sides are used only for very small 
reservoirs and need not be discussed here ; for larger sizes the 
walls are made cylindrical in order better to resist the internal 
pressure. Cylindrical cast iron tanks can be advantageously 

1 



;"""' 




^ 




-r— 


' — 


- 


1 1 

^^ — ^; 


^==:;= 


RLJ 


ti 




=-^ 





_.. 


. ■ .^^r^z^- 






— ZL 


==^a 


== ^ 


::=-- — =^ -- — ^- 




[J: ----- - 


-^ 


= 




■ ^^|T.r=l^ 


= == 





=— 







I 


^^- 




r^=-- •; . 


= 


A. 

3 


_;..-_. 


Tl— 


Hr 


^2ia=^- 


II 




__iL 


1- - 1- 



T 



T 



Fig. 1097. 



used for water up to 1000 cubic feet capacity. A good construc- 
tion has already been shown iu Chapter IV, as made by Lauch- 
hammer's Iron Works, of Groditz, and used iu many places. 



Fig. 1097 shows a tank of this sort. The water is delivered 
at E ; A, is the discharge, and 6'^ the overflow. The thickness 
of the walls is made about j^^-iuch ; the fiat bottom rests on a 
strong fioor of wood carried by heavy beams. The fiange 
joints are made as iu Fig. 26S, 269. If //, is the greates head of 
water in the tank, the pressure per square inch on the bottom 
will be/ = 0.434 h, h being taken in feet, and we have for the 
thickness.^", when D is the inside diameter according to (324) : 







I 
2 


P 

s 


= 


0434 

2S 


h 


E. 

71* 


ramplc.—li & = 
118X9-83 


0.25 in 






118 ius. 
h i.^ Fine 


h 
I1 



, — = 0.217- 



0.25 



(367) 



" 5 



1C09 lbs., whicli is such a moderate value that the tank is 



amply secure. If we take the diameter of the bolts at ^^-in. for the joint 4 
inches deep at the bottom of the tank, and let n, be the number of bolts, 
and further put the permissible load upon each bolt at 275 pounds, wc 
have : 

4 Z> X 0-434 A = 2 ?/ X 275 

r , ■ , 4 X 118 X 0.434 X 9-83 ^ I.- 1 - r t\ J^- ^ 

from which n — r^ — ^— ^ = 3.6 which gives for the distauce 

2 X 275 



from centre to centre of bolts, 



3-6 



= I.I I in. or about i^^ ins. For the joint 



half-way between the top and bottom of the tank the pressure would be but 
half that at the bottom and the bolts may he spaced proportionately wider, 
say about 2 inches apart. The total contents of the tank will be — 742 cubic 
feel = 5550 gallons. 

In using cast iron tanks of this sort care must be taken to 
avoid filling them with licmids which have an injurious action 
upon the rubber packing of the joints. 

i 355- 
Riveted Tanks. ..^ 

When tanks of large capacity are required, wrought iron or 
steel must be used in their construction and these involve the 
use of riveted joints. With tanks of large diameter construc- 
tive difficulties arise in connection with the flat bottoms. 

In the United States, oil tanks are made with flat bottoms, 
carefully bedded iu cement, and similar tanks are used in Ger- 
many for water. It is, however, found that greater facility of 
construction, as well as economy of material, is obtained by 
making the bottom convex, as will be shown. 

A very frequent and useful form is that in which the bottom 

ti 




:Fig. 109S.", 

is made iu the shape of a spherical segment. Fig. X098 a, the 
tank being supported on a flanged ring riveted to its circumfer- 
ence and the ring standing on a support of masonry. 

The construction of the supporting ring is shown in Fig. 
109S d, from the design of Prof. Intze. 

The tension in the inclined direction of the bottom of the 
tank is carried by the lower half of the supporting riug, while 
the upper portion is siibjected to the pressure of the tank at 
right angles to the vertical. This latter force is well resisted 
by a ring of angle iron running entirely around the tank. 

The calculation of the bottom of spherical segment shape is 
as follows : 

If J? is the radius of the sphere of which the segment is a 
part, we have from ^ 19, Case II. : 






p_ 
■2S, 



in which d^ is the thickness and ^i the stress therein due to the 
pressure/. The pressure is the greatest at the lowest point of 
the bottom where the height iu feet of the column of liquid is 



THE CONSTRUCTOR. 



261 



equal to A, so that if ff, is the weight of a cubic inch of the 
liquid^ = 12 /i a. We then have: 



J? 



1 2/1 a h 

2 S^ 5i 



which for water gives, a = 0.0361 
R 



(368) 



= 0.217 



h 
5^1 



At each higher point of the bottom the pressure is less, until 
at the edge of the bottom the height /;, is diminished by the 
depthy, of the bottom. For simplicit}', however, it is custom- 
ary to make the entire bottom of the same thickness <S, which 
is required for the lowest point. 

For the thickness of the cylindrical walls of the tank at the 
bottom we have the pressure/) = 0.036 (A — ■/) both h and_/, 
being in inches, and from (367) 



6 

15 



2 ■ 5 ~ ' S 



this gives S in feet, hence we have for & in inches : 

cJ = 12 X 0.217 D — =— = 2.604 D —~^ ■ (369) 

In order to obtain good proportions it should be considered 

that as /? diminishes, the ratio of — becomes smaller, while as 

£> increases the size and thickness of the bottom increases. 

An approximate formula by which the minimum amount of 
material will be required is : 



D 



•366^ 



Q 



(370) 



in which O is the volume of the material iu cubic feet to be 
contained in the tank. 

For the height /I of the wetted portion of the surface we 
have : 



// + — = // 
2 



2 



£> 



(371) 



if we assume, as we may with sufficiently close approximation, 
the segment of the sphere to be practically that of a paraboloid. 
The same remark about the most economical ratio of depth 
to diameter applies here as in the uote to ^ 354. 

Example I. — For Q = 47,000 cubic feet we have from (370); 



[.366 -^ . 



47-36- 



A carefully calculated tank at Halle, of this capacity (1200 cu. metres) was 
made 51.88 feet diameter. 

ir O = 65,60-1 cu. ft. xve have D = i.366^7o,ojo = 56.3 ft., while a jank of 
the same capacity at Essen is 5S feet in diara,eter. 
The water towir at Neustassfurt has a capacity Q = 21,160 cu. ft., and is 



39.36 ft. diameter ; according to (170) it would he D = 1.366^21,160 = 37,79 ft. 
All three cases thus agree well with the formula. 

For the depth yj of the concave bottom, we have for any given 
radius J^, the expression 



2 J?/-/' 



\ D\ 



from which we get 



B ^ \D J 4 



(372) 



It is found convenient, but not essential, to choose such a 
value for R, that t\ = i\ when 5 =: S^. To accomplish this re- 
sult, the conditions which obtain for the equations both for (S, 
and (5 must be fulfilled. These are : 



R_ 
D 



in/ 

h ' 



(■hence, -— 



D 



R_ 



(373) 



The following table gives a series of numerical values for 
these relations : 



R ! 


0-55 


0.60 


0.625 


0.65 


0.70 


0-75 


0.80 


0.85 


0.90 


0.95 


1.0 




/ 

D 


05 


0.32 


C.7 


0.2s 


0.23 


0.21 


0.19 


0.18 


0.16 


"■--i 


0.14 


0.134 




h 
D 


I.O 


0.71 


0.68 


0.67 


0.66 


0.70 


0.76 


0.88 


1.07 


1.52 


2.84 


00 


h_ 


-o.sf __ 
D 


0.75 


0-55 


0.54 


0.54 


0.56 


0.59 


0.67 


0.79 


0.99 


1-45 


2.77 


CO 


i-( 


^)- = 


0.17 


0.07 


0.05 


0.04 


0.03 


0.03 


0.02 


0.02 


0.02 


0.02 


0.01 


O.OI 



These relations are also shown graphically in Fig. 1099, and 
the results are interesting. It will be seen that in order to have 
(!j = i when Sj := .S we must always make R <. D. It also 
appears that the best ratio of depth to diameter occurs when 

3D 

-— is about equal to 0.60, for then /; — ■ 0.5 /nearly approaches 

0.5 D ; this, however, is only approximate. It thus appears 
that the two conditions of greatest economy of material and 
equality of value c!j and t', cannot be attained at the same time 



o.c^'> 




0,85 



0,79 
o,6r. 

11,60 
o,5.'> 



,00 
aao 

0,89 

0,75 

<l,70 
0,65 
06'23 
o.eo 

0.5% 



Fig. 1099. 

exactly. The most useful ratio in practice will be obtained by 
selecting a value for /?, according to (370). 

The value R = 0.5 D, which corresponds to a hemispherical 
bottom, is useful to the extent that when the supporting ring 
is placed at its upper edge there is no lateral pressure produced 
tending to compress the ring, as there is in all of the other 
cases. The hemispherical bottom, however, offers too many 
constructive difficulties to be much used. 

Example 2. — Let Q = 53,000 cubic feet. We have from (370) ; 
D = i.336\?/ 53,000 = 50-28 feet and according to C371), A '•$/= 0.5 Z> == 

25.14 ft., and combiuing these again we get: (7=25,14X0.7854(50.28)2 = 
49 420 cu. ft., which is a little under the required content, but shows the cor- 
rectness of the proportions. 



262 



THE CONSTRUCTOR. 



If we now make /= o 21 Z* = 0.21 X 50.28 = 10.56 we have from the above 
table, i? = 0.7 Z> = 0.7 X 50.28 = 35.2 ft. We have from (371) h = 0.5 Z? + 
0.5/"= 0.605 D = 30.42 It. The height of the wetted perimeter will be // = 
/i — /= {0.605 — o-2t) -^ = 0.395 D = 19.86 ft. 

Taking for the stress in the metal at the lowest part of the walls of the 
tank wc have from (369) : 



5 = 2.604/? 



H 



ir- 

2.604 X 0.395 —^ = 0.372 m 



For the bottom we have 



j.6o4 J? ' 



0.7 X 2.604 X 0.605 ■ 



7^2 



: 0.4 m. 



and — ^ 1.07 ; that is, the thickness of the bottom is 7 per cent, greater than 

o 

that of the lowest row of plates in the walls of the tank. 

If we make the tank with six rings of 3 ft. width and one of 2 ft. we get for 
the thicknesses: 



Depth = 

5 
Calculated = 

6 
In practice = 


19.86 


16.86 


13.86 


J0.86 


7.S6 


4.S6 


1.86 


0.372 


0-315 


0.260 


0.203 


0.147 


0.091 


0.035 


%" 


T5 


w 


'A." 


Ji" 


J4" 


M"- 



The latter figures show an excess over the theoretical thickness, but the 
excess is needed for stiffness and for constructive reasons. The thickness 
of the bottom, as already calculated is 0.4 in., but in practice would prob- 
ablv be made y'g". 

The riveting may be made the same as ordinary boiler riveting ; and from 
the table in g 59, we find for 5 = ^-s", d= \\" and for single riveting the 

modulus of efficiency is 0.47. This gives a stress of -^-— = 15,000 pounds, 

0.47 

which seems rather too high. For this reason the two lower seams at least 

should be made with double riveting ; which gives a stress of ^ = 11,800 

0-59 
pounds. The seams of the bottom should always be made double riveted. 

Example 3. — Let Q again be taken as 53,000 lbs. "We will now proportion 
the tank so that 61 = 5, and take i? = 50 ft. 

R 



In order that l\ shall at least equal 5, we will take 



D 



0.625 whence f 

f = 



= 0.25 D = 12.5 ft. We then have /i = 0.67 D = 33.5 ft., and h — 

(0,67 — 0.125) D = 0.545 D == 27.25 ft. We therefore have 

Q = 0.7854 X 27-25 X (50)- = 53.500 cu. ft. 

Tphich agrees quite closely enough with the original assumed capacity, 
//will be = to /i — /={o.6j — 0-25) D = 0.42 D = -21 ft. 
We therefore have for ihe lowest cylindrical portion of the tank : 



6 = 2.604 X 0.42 
and for the bottom : 

0,625 X 0.67/72 



D^ - 502 

—=- = 2.004 X 0.42 = 0.3906'' 

o 7000 



01 ■■ 



- 2,604 ■ 



^0.625 X 0.67 X (50)^^ 



== 0.3894" 



thus giving practically 5 = ^1. 

The tank will be heavier than the preceding proportions give, as might 
be expected, but the excess weight will be only about i per cent. 



§ 356. 

T.^NKS WITH Concave Bottoms. 

The question of the action of the forces upon the bottom of 
a tank as discussed in the preceding section, was first 
thoroughly investigated by Prof Intze, whose valuable re- 
searches have practically revolutionized the construction of 
riveted tanks.* The following discussion is based on Intze's, 
but the calculations are simplified and abridged. 

Fig. 1 100 shows two forms in which the spherical segment 
may be used, a, with convex or hanging bottom, as already dis- 
cussed, and b, with concave or reversed bottom. In both forms 
the pressure of water on the bottom produces a stress at the 
base of the cylindrical portion of the tank in the direction of 
the tangent to the curve of the bottom, the stress acting in- 
wards in case a, and outward in case b. It is desirable to make 
the construction such that this force is received by the base 
ring and not by the shell of the tank. In every case, however, 
an increase is required in the thickness of the bottom of the 
tank. 

There is also a force ;•, acting at right angles to the tangent 
or normal to the curve of the bottom of the tank, and the deter- 
mination of both of these forces is a matter of importance. 

If G be the weight of the liquid, and a the angle which the 
tangents make with the axis we have for case a, for the two 



* See the article by Dr. Forchheimer : " On the Construction of Iron 
Tanks, for Water, Oil and Gas, according to the Calculations and System of 
Prof Intze, of Aachen." Schilling's Journal iiir Gas-beleuchtung, 1S84, p. 
705. 



lateral forces which act, each on one half the circutafe 
the base ring of the tank : 




Ai 



1 


'Vf:' 




i. \ 

t! 


i ..;■ 

' \ 1 / 

\ 1 / 





Fig. II 00. 

2 cos a 
producing a load Si per running foot : 

5 



■s, =■ 



-£) 



D'+f 



Substituting for G, its value 

E'r „ 

in which y is the weight of a cubic foot of the liquid, we get : 

^ r, /■ . 2 



-/) + ^/ 



•)] 



'4["-^H^(*)-] 



In this h is the distance from the level of the surface of the 
liquid to the crown of the curve of the bottom, and for the case 
b, we have : 

The last member in the brackets is always very small in value 
as will be seen by reference to the table in the preceding sec- 
tion. It can therefore generally be neglected, when we have 
for both cases : 



s — y ■ 



J? 



i) 



(374) 



The detailed determination of the forces /; and 4, need not 
be gone into here, we have for both cases : 

i = yJ?(k^/)-s = y^(A^±/) . . . (375) 

There is also a third force 71, acting upon the rim of the 
spherical bottom in the direction of a great circle at right 
angles to the plane of the drawing, for which we have per run- 
ning foot : 






(376) 



and finally for the crown of the curve, where the force ii in a 
great circle is : 

"o = '■ T '^ (377) 

These formulte will be somewhat simplified if we take the 
height H, of the wetted portion of the cylinder, whence 
h^H±/. This gives: 



u^y^H, u, = y^{H±/) 



■ -(378) 



THE CONSTRUCTOR. 



263 



These are the necessary formulas for the calculations of 
spherical bottoms. The following points are to be noted : 

I. For the convex bottom (Form, a) u„ has the greatest value, 
that is, the stress must be calculated for the deepest point if (S„ 
is to remain constant; 2. For the concave bottom (Form. ^) t 
has the greatest value, and must be used to determine dj ; 3. 
The supporting rim should be capable of sustaining s, if the 
shell is to be free from any stress due to the bottom of the 
tank. 

The determination of (S, is the same as before. 

If we divide the values for «<, and ;', by 12, we get the stress 
per running inch, and by using the weight cr of a cubic inch of 
the liquid and taking R in inches, we have for the convex 
bottom : 



<5, 12 /;/ G 
R ^ 2 5i 


= 


12 


2 5, 


oncave bottom : 








(Ij 12 /;' a 
R 2S, 


12 


G 


2 5, 



(379) 



/ 



• (3S0) 




[Fig. iioi. 

If the bottom is made conical, projecting either within or 
without as in Fig. iioi, the height of the cone being/", we have 
for the weight of the body of liquid • 

and taking the component as before in the direction of the 
angle of the cone, we have : 



G 



5=— D s = 

2 2 cos a 



■whence : 



5 = r — Z)2 

4 TT D 2 cos 



r(--i) 






D 



2 2 cos 



.(--i) 



2 

But IS equal to the radius R oi a. sphere inscribed within 

cos a 

the cone ; whence we have : 
R 



s=y 



C-'i) 



We also have for i the same value as for u, and 



R 



t = u — } —H 



(381) 



(382) 



For the inverted hanging cone bottom, form c, the greatest 
of the three forces is s, while for form d, in which the cone pro- 
jects into the tank t = u, is the greatest, and we use in practice 
for form c; 



and for form d : 



R 

A 

R 



2 5, 



}2.a_H_ 

2 J 



(383) 



(384) 



The conical form of bottom, as will be found upon compari- 
son, requires about 40 per cent, more material than the spheri- 
cal, but as will be seen, its use under some circumstances is 
advisable. 

Instead of using a complete tone, the bottom may be made a 
truncated cone, the tank being formed of two concentric cylin- 
ders connected by a ring-shaped bottom, as in Fig. 1102. 



V 



•D6 - 

..-D-l-... 



f 

i 



;<-D 



^ 



.--jt:-.. 



H^ 



% 





i, 

,-' i \^ 


g 


^,-=^ 




^ 



H 




Fig. 1102. 

These may be made either projecting inward or outward. 
Following the same line of investigation as in the previous 
cases we have for case e : 



7 - (/?o^ - n^: 
4 



1/ 



4 



B) H{D + 



i^o - D) ) 



and for casey.- 



This gives for case e 

R f r ^i?, 



'"2 ] I \D 
and for case,/ 
D R 



-I \H- 



Y 



D, 






3 



c 



] 



H ■ 






(385) 



D 



^■] 



(38h) 



in which R is the radius of the sphere inscribed within the 
truncated cone.* The forces / and u are obtained in a similar 
manner as before. 

The subject of truncated conical bottoms will be discussed 
again. 

We have for the weight (t, of a cubic inch of various liquids : 

Water 0.0361 lbs. 

Petroleum 0.02S9 lbs. 

Linseed Oil, at 12° C. = 54° F o.o339 lbs. 

Bisulphide of Carbon, at 0° C. == 32° F. 0.0459 lbs. 

Glycerine, at 0° C. = 32° F 0.0455 lbs. 

Beer, at 0° C. = 32° F 0.0372 lbs. 

Alcohol (absolute), at 20° C. = 68° F. . 0.0286 lbs. 

In the construction of tanks, it is necessary also to consider 
the peculiar properties of the various liquids. For alcohol no 
packing should be used in the joints, the tightness only being 
secured by caulking the riveted seams 



* \i Dq be made o, the formulae will become those for complete cones, as 
indicated in the dotted lines. The formulae for the weight might also be 
symmetrically expressed: the form used has been selected because it makes 
H the higher of the two walls, which is more convenient in numerical cal- 
culation. 



264 



THE CONSTRUCTOR. 



I 357- 
Combination Forms for Tanks. 

In the forms of tanks already described the force 5 sin a 
acts either to press the supporting ring inward or outward in a 
direction radical to the axis, according as the forms a, c, e, or 
b, d, f, are used. This circumstance lends itself very fortu- 
nately to Prof. Intze's method of construction, since by com- 
bining both forms iu one bottom the forces may be made to 
equilibrate each other and thus relieve the supporting ring from 
all radial stresses. 




Fig. 1103. 

This idea may be carried out in many ways, as by combining 
forms d and/, Fig. 1 103 a, or forms e and b, Fig. 1 103 b. or using 
all three forms as iu Fig. 1103 c, the inner vertical walls being, 
in these combination forms omitted.* 

The forms shown in the illustration also have the advantage 
of reducing the diameter of the supporting ring and hence re- 
quiring less extensive foundation walls. 

In order that the supporting ring may be free from radial 
stresses, the condition ; 



s' sm 0.' — s" sm a' 



O . 



.... (3S7) 

must be satisfied. This simple equation cannot be briefly 
solved numerically, hence an example is here given of its 
application. 

Exa7nple.—0\veii a water tank of the form and dimensions of Fig. 1104, th 
radius of curvature of the bottom being R". The first member of the equa 
tion belongs to the outer, and the second to the inner portion of the tank 




Fig. 1 104. 

For the first member we .have for s', from (385) ; D^ = 12, D = 4cy H = 6,/ = 

2.4, whence tan « = — ^ =1.667 = tan 59°. This gives sin « ' = 0.8572, and 

2.4 

cos a' ^ 0.5150, and 

R. „ °-^ = ^_ = 3,8f3 and 
Sin a o 515- 

.'Sina= 0.S37. V :<o.5Z)(^){ [(^r)^/^- 
(^-^)(8X6-o.8X.4) = 



{y X 0.5 D) 0.8572 X o-:23 (48— II. ^y — 
(V X 0.5 D) 0.2769 X 36-8 = 10.19 (V X 0.5 D). 
For the second member we have from formula (378) ; 

s" sin a.'' = sin a" y 0.5 R" {H — 0.$/") 
in which both R" and a" are unknown, hence we introduce ^" and have: 

s" sin a" = y cos ^" R" (3 — 0.25 R" (i — cos ^") ). 
Introducing these into the equation of condition, we get: 

10.19 X 0,5 Dy — cos ^" R" (3 — 0.25 R" (i — cos ^") ) y = o 

0.5 D 
But ■ sin |S whence : 

10.19 

We may obtain a first approximation for &" by neglecting the second 

member of the numerator. This gives tan ^" = = 0.2954 = tan 16'^ 25'. 

The true value must be somewhat less. Assuming it to be j3" = :6° 20', the 
tangent = 0.2930, the sine = 0.2812, the cosine = 0.9596. We then have R" =^ 

0.5 D 



= 7. II ft. and 10.19 X 0.2930 — (3 — 0.25 X 7." X 0.0404) = 



sin /3" o 2I 

nearly. Numerically this gives; 

2.9S6 • — 2.92I 



: 0.05S 



* Combination forms of this sort have been patented by Prof Intze, (Ger- 
man Patents, No. 23,187, 24,951 and built for oil, water, gas, etc., at the works 
of F. A. Neumann, at -iacheu. 



or, since the weight of a cubic foot of water = 62.4 lbs., the unbalanced 
radial force upon the ring is 62.4 X 0.058 ^ 3.62 lbs. per running foot, which 
is so small as to be unimportant. 

The question may properly be asked, as to the stresses upon the support- 
ing ring when the tank is not full, that is, when //varies. In answer, it is 
true that the pressure on the ring necessarily changes. Suppose jy= 3 ft. 
We then have for the first member of the equation ; 

y '-' 0.5 D X 0.2769 (24 — 11.2) = y ;< 2 X 0.2769 X 12-8 = 7.088 y 

and for the second member 

7 :< 0-9596 X 7-II K^-S — 0-25 X 7-11 X 0.404) = 6.823 X 1.428 y = 9.743 V- 

"his gives a pressure of 

7.088 y — 9.743 y = — 2.655 \ 

or 2.655 X 62.4 = 165.67 lbs. per running foot acting from without inwards, 
which is large enough to be worth considering. It i^therefore important to 
base the calculation upon a depth of water which will be usually main- 
tained in the tank. The propoitions may also be so made that the forces 
will be in equilibrium when the tank is half full, when a greater depth will 
cause an outward pressure and a lesser depth an inward pressure. 

Tanks constructed on the combination are well adapted for 
use with gasholders, the level of the water remaining so nearly 
uniform that the supporting ring may be kept free from any 
lateral pressure. 

1 358. 

High Pres.surk Reservoirs or Accumui,ators. 

The forms of tanks already described are intended to be 
placed at such elevation either in buildings, or towers or on 
natural elevations that the liquid is delivered through pipes at 
the desired pressure. 

In this way a water tank with a pump and the necessary pip- 
ing forms a storage system, an overflow being provided as a 
security against flooding the tank. Systems of oil storage are 
constructed also in this manner ; and on a small scale the water 
tank stations for railway service come under the same classifi- 
cation. These water stations are usually provided with steam 
pumps, although windmills are often used, especially in the 
United States. 

It is a question whether the required pressure might not be 
obtained by the use of compressed air, the tank being closed at 
the top and the confined air exerting by its elasticity sufficient 
pressure to obviate the necessity of elevating the tank upon a 
tower to obtain the necessary pressure. 

For high pressure water systems for operating hydraulic 
machinery the use of weighted devices, as suggested long since 
by Armstrong, has superseded ihe open water column, such 
devices being generally known as Accumulators. 

The volume of such accumulators is generally quite small, 
but the pumping mechanism is so efficiently devised as to ena- 
ble them to possess a very extensive capacity. The pressure is 
obtained by means of a weighted plunger, the overflow being 
replaced by a safety valve. 

Fig. 1 1 05 shows an accumulator built by C. Hoppe, Berlin.* 
This is weighted to a pressure of 20 atmospheres, or nearly 300 
pounds per square inch. The plunger is 1734^ in. diameter (450 
mm.) weighted with shot which is enclosed in a cj-linder. The 
plunger is shown in the highest position. When it reaches the 
position the lever and connections ]\J M' act to shut off the 
steam from the duplex pump, and at the same time the rod .f 



* All dimensions in the illustration are in millimetres. 



THE CONSTRUCTOR. 



265 



relieves the safety valve. When the use of the water causes the 
plunger to sink, the steam is turned on and the pump starts. 
If the pressure should be suddenly released by the bursting of 



the efforts to attain compactness, having led to a vast number 
of modifications of the original simple forms. 
The various boilers used in Germany may be reduced to 




Fig. 1 105. 

a pipe, the sudden drop is received by heavy beams, and at the 
same time the stop P' strikes the lever /'and checks the water 
flow in time to moderate the shock. 

An accumulator for very high pressures is shown in Fig. 
1106.* This is designed by Tweddell for use for operating riv- 
eting machines, punches and similar tools. The plunger c, is 
stationary ; the cylinder rf, sliding upon it, weighted with rings 
rfi of cast iron. In the lowest position the cylinder rests upon 
vertical buffers of oak. The water is delivered under high pres- 
sure at //, while the w'ater is taken off for use through suitable 
valve gear at A ; the safety valve is at b'. The plunger is of 
the differential variety similar to those shown in Fig. 977 b, and 
Fig. 981 b. The difference in diameter between the two por- 
tions of the plunger is the space to be filled by the entering 
water, the small annular area bearing the total weight, thus 
giving a very high pressure per square inch. The pressure at- 
tained when the cylinder is stationary is about 100 atmospheres 
(1420 pounds), but experimental investigation has shown that 
when the weighted cylinder is permitted to descend rapidly the 
pressure reaches as high as 193 atmospheres, (2740 pounds), so 
that it is worthy of note that the attainable water pressure in 
such devices may reach double the statical pressure. 



? 359- 
StB-A-M Boilers, Various Forms. 

Steam boilers may properly bejconsidered as reservoirs for 
vapor of water, while at the same time they serve as generatois 
of force by the application of heat. The pressure is produced 
by the heat, the feed is effected either by a pump, as Fig. 975 d, 
or injector, Fig. 971. The overflow is represented by the safety 
valve, and the observation of the water level is provided for in 
a variety of ways. 

The forms used for steam boilers are very numerous ; the 
great variations of size, the varying conditions of locality, and 




Fig. 1 106. 

eight principal classes, examples of which will here be 
given. 

I. Plain cylinder boiler. Fig. 1107, usually placed in the hori- 




FiG. 1107. 

zontal position, and now principally used in iron works where 
the waste gases from the furnaces are used. 





* See Proc. Inst. C. E. Vol. LXXIII. 1883, p. 9=. 



FiG.EoS.t 

2. Cylinder Boiler with Heater, Fig. 1 108. The cylinder has 



266 



THE CONSTRUCTOR. 



added to it a "heater " or auxiliary cylinder placed beneath and 
forming a part of the boiler, being entirely filled with water 
and surrounded by heated gases. Besides the usual form, there 
are Henschel's in which the heater is placed at right angles to 
the main boiler, and the vertical form. 



tube to produce a circulation by the difference of temperature 
between the inner and outer tubes. Boilers with large and 
small cross water tubes are shown at b and c. 

A ninth group might be formed of special combinations of 
the eight groups above shown. 




Fig. 1 109. 

3. Tubulous Boilers, Fig. 1109. This class includes boilers 
made of tubes 6 inches in diameter and under. Of the arrange- 
ment shown, a, is Belleville's ; b. Root's ; and c, Howard's. 




Fig. 1 1 10. 

4. Flue Boilers, Fig. mo. Flue boilers are constructed with 
internal f "■es entirely surrounded by water and containing the 
furnaces, fire and heated gases. The Cornish 
boiler, a, is made with a single flue, and the 
Lancashire boiler, 5, with two flues. 

5. Flue Boilers with Cross Tubes, Fig. mi. 
This form, also known as the Galloway boiler 
is constructed with water tubes crossing the 
flvie at various points. 

6. Plain Tubular Boilers, Fig. 1112. These 
are made with tubes of 6 inches or less in 
diameter, through which the heated gases 

pass. The tubes are lap-welded or seamless, and a distinction 
is made between direct and return tubes. 




Fig. im 




Fig. 1 1 12. 



7. Fire Box Tubular Boilers, Fig. 11 13. A fire box consists 
of a box forming a part of the boiler containing the furnace and 





Fig. 1113. 

surrounded with water. These boilers are also made with 
■ direct and return tubes. These are made either with vertical 
tubes as at a, or horizontal tubes b, and are much used for loco- 
motives and portable boilers. 

8. Fire Box Boilers with Water Tubes. Fig. 1114. Of the 
forms shown, a, is a boiler fitted with Field's tubes. These 
tubes are closed at the lower end, each containing a small inuer 





o © 



Fig. 1114. 

In England and America a different classification is made, 
the boilers being divided into two great classes, those which 
consist of a large shell, with the necessary auxiliary parts, and 
those composed of numerous small elements, the number of 
elements being governed by the size of the boiler. These two 
classes are known as ''shell" boilers, and "sectional" boilers. 
The third group shown above, consists of sectional boilers. A 
popular form in many countries is the Harrisou boiler, com- 
posed of small spherical elements of cast iron. The relative 
value between shell and sectional boilers is a question not yet 
entirely settled. The latter form is incapable of destructive ex- 
plosions, such as may occur with shell boilers containing large 
volumes of water. Sectional boilers are also adapted for very 
high steam pressures, but have the defect in many cases of pro- 
ducing moist steam. 

The latest police ordinances in Prussia, which are similar 
to those of Austria, distinguish between "dwarf" boilers 
and ordinary boilers. The former are boilers of small volume, 
less than iS cubic feet ( '< cubic meter) capacity, these together 
with sectional boilers being permitted for small private indus- 
tries. 

§ 360. 

Boii<ER Detaii<s Subjected to Internai, Pressure. 

The walls of steam boilers are subjected to varied and some- 
times complicated stresses greatly dependent upon the method 
of construction. It will only be practicable here to discuss the 
ordinary forms, first taking the parts which have to resist the 
internal pressure. 

a. Cylindrical Details. 

The Prussian ordinance relating to steam boilers, used the 
formula of Bris, for cylinder boilers subjected to internal pres- 
sure : 






D 



000.3 '^ 

e — I 



(388) 



in which i and D are in inches and c is the logarithmic base = 
2.71828, (7 being the pressure in atmospheres. This is closely 
approximated by the simpler formula : 



6 = 0.0015 a D -\-o.i 



(J89) 



The French formula is much the same but gives a slightly 
greater thickness : 



& = 0.0017 (7 Z) -f o.i: 



(390) 



only 73 of this value being used in locomotive practice. On 
account of the large constant added to provide for deterioration 
all three formula must be considered as empirical. At present 
there are nearly everywhere government enactments which 
prescribe the method of determining the thickness of steam 
boilers and regulate by law the limits of construction. 

In most cases the boilers must be subjected to a test pressure 
which may reach double the working pressure. 

The stress existing in the longitudinal seams of a cylindrical 



THE CONSTRUCTOR. 



267 



boiler shell may be obtained with sufficient accuracy from 
(324), as: 



5 = 



2 



D_ 



(39') 



p being the pressure in pounds per square Inch, and D and <J 
being in inches. If we calculate <' from (389) and determine S 
from (391) we have the following results 



a = 


1 1 
4 = 60 lbs. 


7 = 105 lbs. 


10= I 


1 
50 lbs. 


13=1 


7.S lbs. 


D 


J 


5 


& 


5 


(5 


5 


(! 


5 


24 
36 
42 

72 


0.24 
0.31 
0.35 
0-43 


3000 
3500 
3600 
5000 


0-35 
0.4S 
0.54 
0.S5 


3600 
3900 
4000 i 
4400 j 


0.4S 
0.64 

0.73 
i.iS 


3750 
4000 
4300 
4600 


0.58 
0.80 
0.92 
1-50 


37°o 
4000 
4000 
4200 



This table shows that the formula gives for large diameters 
and heavy pressures, thicknesses which are excessive, and with 
quite moderate stresses. The stress at the riveted seam will be 
greater, and from \ 59 we have for the stress in the perforated 
plate : 



for single riveting; S' ^= 



S 



for double riveting S', 



5^ 



(392) 



in which, if d, is the diameter of rivets and a, the pitch, 

a — d a, — d 

(i' = and li-'o -= . 

a ' ' (?2 

Even with this increase the stresses fall below the values 
which good boiler plate should properly bear. In practice, 
smaller values are often used for S than are given by (389) 
especially since mild steel came into use for boiler plate. 

The stress which comes upon the rivets, according to \ 59, is 
greater than that upon the perforated portion of the plate. 
This, however, should be considered in connection with the 
fact that rivets are generally made of a still better grade of iron 
than the plates. 

At the present time the disposition is apparent to break loose 
from set rules for the thickness of boiler shells. Careful 
designers aim more and more to investigate each case for itself 
and endeavor to adapt both design and material so as to obtain 
at the same time the greatest strength and economy. The more 
recently designed ocean steamers are fitted with boilers as 
large as to feet in diameter, operated at pressure from 160 to 
250 pounds pressure. The older formulce cannot be used for 
such e.-s:treme values, and every resource of the art must be used 
to reinforce the strength of the plates and the riveting. The 
method of group riveting, {\ 57) is here found of value and is 
already used to some extent. 

Longitudinal Seams. 

For all large steam boilers the longitudinal seams are double 
riveted. For plates y\" thick and over a modulus 9./ = 0.76 to 
0,73 is obtained which corresponds to a ratio of S'„ : S of 1.32 to 
1.37. It is more and more made a point of importance that 
these joints shall not be exposed to the direct action of the 
fire. 

A construction especially intended to meet this point is 
shown in Fig. 1115 in which the entire shell is made of two 
sheets, the lower sheet comprising obout f of the entire cir- 
cumference.* 

.Another method which bids fair to become very important, 
is to weld the longitudinal seams, this being more and more 
and more used for large boilers. The welding is accomplished 



* Boilers of this sort have been made at the Erie City Iron Works. Erie, 
Pa. See Trans. Am. Soc. Mech. Engrs. Vol. VI., 1SS4-5, p. no. SchefHer, A 
New Method of Constructing Horizontal Tubular Boilers. The first boiler 
was 16 ft. by 60 in., Y?. thick, of mild steel, 60,000 lbs. ultimate strength, 
30,000 lbs. proof strength. 



either by furnace heat or by water-gas burners, or as= more re- 
cently by electric welding by the Bernados' method. 







Fig. IU5. 

In Fig. 1 116 is shown the cross section of a marine boiler, 
constructed by H. C. Stiilkeu, of Hamburg, the two longitudi- 
nal seams being welded.-- Both seams are reinforced by double 
riveted flaps the strength of the plates being reduced by the 
rivet holes. 

The joint, however, is preferable to a lap joint and needs no 
strengthening. The pressure in this boiler is iSo pounds. The 




Fig. II 16. 

strength may be calculated as follows ; the diameter being 76.5 
ins. and the plates Ys, in. thick. From (391) we have : S =: 

iSo_xj6^5 
2 X 0.S75 
flaps the pitch <;., is 2.9 ins. and rivet diameter d 



7S6S pounds. For the double riveting in both 

i in. This 



gives for the modulus of efficiency ii'., = 



29^_aS75 
2.9 



= c.7 and 



the stress in the perforated plate in the longitudinal seams is 
7868 



0.7 



= 11,240 lbs. The thickness of plates according to for- 



mula (389) ■\vc.uld be 0.0015 X 12 X 76.5 + o.i = 1.5 ins., in- 
stead of Ys, ins. 

A third method of construction which may become impor- 
tant is to construct the shell in a single piece of mild steel by 
the Maunesmann process of rolling. This method would be best 
of all, since the question of the strength of the riveted seam 
would be entire!}- eliminated, and the high elastic limit of the 
material would permit correspondingly high working stresses. 
At the present time, however, the JIannesmanu rolling mills 
cannot make tubes over 24 inches in diameter. 



* See Zeitschr, D. Ing., 1SS6, p. 109. 



268 



THE CONSTRUCTOR. 



Circui>iferential Seams. 
The cross section of the boiler shell, when the head is fast to 

it, is subjected to a force — £>'/> = S.^ tv D i', in which S^ = 
4 

_4 ; that is half as great as the stress S, in the longitudinal 

<S 
seams. For this reason it is deemed necessary to use only 
single riveting for the circumferential seams. It will also be 
shown hereafter, that the cross section of the shell can be re- 
lieved of this load. 

Openings in the Shell. 

The openings for the steam dome and manholes weaken the 
boiler, and in some instances explosions have been caused by 
cracks radiating from such openings. All such openings should 
be carefully reinforced by riveting on rings of wrought iron or 
preferably steel, as shown hereafter in Fig. liiS. The size of a 
manhole opening should be about 12 by 18 inches, and when 
practicable the short axis of the oval should be placed length- 
wise of the boilers. 

b. Spherical Details. 

A sphere of the diameter D^ with an internal pressure^, will 

be subjected to a force — D^- p, which is the same as already 

found for the cross section of a cylinder, and one-half that on 
the longitudinal seams. The thickness, therefore, need be only 
half so great as that of a cylindrical shell of the same diameter, 
i. e., D = D.^ If, however, both vessels are to have the same 
content we must have D^ y> D. If the cylindrical shell is made 



with flat heads its content will be — Z)^ Z, 

4 



Z)3 



G). 



and the spherical vessel will have a content = -— D^ ; hence 

5 



we must have i3°i = — V^ 



D 



)■ 



For the thickness of metal we have : 



— Dp — 



D,p 



and for the respective surfaces : 



F-- 



D L-\ D\ and F^ ■■ 



D\. 



Assuming the heads of the cylindrical vessel to be made the 
same strength as the shell, we have for the material required 
for each case : 



E- (5 — n^ ~ 

4 -ii 

Making 5 =:= Si and putting for D^^ its value — /)3 ( 75- ) 
get : 





L 


F,6, 


3 D 


F6 


4 J^ , I 




z^+.- 



(393) 



for the ratio between the amount of material required for 
spherical and cylindrical vessels. We have for : 

-^ = I iK 2 3 4 5 6 00 

F6 ~ 



0.50 0.56 0.60 0.64 0.67 0.68 0.70 0.75 



showing that the spherical vessel is in all cases the lighter 
form. 

The earliest boilers were made in the spherical form, but 



soon abandoned on account of the demand for increased heat" 
ing surface and small content. The spherical form is, however, 
well adapted for units for sectional boilers.* 

For spherical ends of cylinder boilers, as in Fig. 1 107, and for 
the heads of domes, and auxiliary drums, we have for the thick- 
ness, R-^ being the radius of the sphere : 



2 5 



(394) 



which gives, when .S, = 5 the same value for the thickness d, 
as in the shell when R^ = D. This latter condition cannot 
always be fulfilled since the curvature of the boiler head is 
usually controlled by the dies with which the press is provided. 




The head is usually joined to the shell by being flanged or 
turned over around the edge in the flanging press, thus enab- 
ling a joint to be made as at a. Fig. 11 17 ; or it may be made 
with a ring of angle iron, as at b. Here the circumferential 
force, as considered in § 355, may be taken into consideration, 
especially the radial component s sin o, since this acts to draw 
the shell inward. It is, however, hardly necessary to take this 
into account as the flange of the head reinforces the shell amply 
at this point. . 

c. Flat Surfaces. 

Unstayed flat surfaces can onh' be u.sed in boilers of small 
dimensions, as already shown in § 19, and should only be used 
for heads of steam domes, auxiliary heaters, and the like. 
Where extended flat surfaces are used, it is necessary to adopt 
some method of staying ; or in other words to subdivide the ex- 
tended surface into supported portions small enougli to be of 
ample strength and at the same time of moderate thickness. 

A number of methods of staying flat surfaces are in practical 
use, those most generally employed being shown in Fig. iiiS. 




Fig. iriS. 

Stay bolts, such as shown in Fig. iiiS u, (see also | 61) are 
used for parallel surfaces which are near to each other. Those 
shown at a are made with nuts instead of riveting the heads 
as is sometimes done. Flat surfaces which are farther apart 
are secured b}* anchor bolts, as shown at b ; these are practi- 



* The Harrison Boiler, the pioneer of modern sectional boilers, is coni- 
posed of spherical units. Trans. 



THE CONSTRUCTOR. 



269 



cally long stay bolts. These are shown reinforced by large 
riveted washers under the nuts. 

Stay bars, as shown at c, are used for staying crown sheets of 
fire boxes in marine and locomotive boilers. Stay tubes, such 
as shown at d, are used to strengthen tube sheets. These are 
heating tubes about '4 to -j% in. thick reinforced at the ends and 
screwed into the tube sheets. Gusset plates c, Fig. b, are used 
to stay flat heads to the shell, and are used Ijoth in land and 
marine boilers.* 

I 361. 

BonER Flues Subjected to External Pressure. 

The stresses which appear in the case of a boiler flue subject- 
ed to external pressure are similar to buckling stresses upon 
columns, rods, etc., since beyond a certain increase in pressure 
when a slight departure from the true cylindrical form occurs 
a sudden collapse follows. The smaller sizes of flues used in 
the ordinary tubular boiler possess ample strength against col- 
lapsing, but for larger flues such as are used in Cornish and 
Lancashire boilers the question of strength to resist collapsing 
must be considered. The experiments of Fairbairn have demon- 
strated that the length of the flue has an important influence 
upon the resistance to collapsing, practicallj' being inversely as 
the length of the flue, or rather as the distance between the 
points at which the flue is reinforced against external pressure. 



stituting these in the formula it will be found if the flue is safe 
against collapsing. 

Exain/^le.— In a Cornish boiler intended to work at 37J: pounds pressure, 
the dimensions are / = 25 ft., Z) = 23 ins., S = 0.25 ins., the flue being made 
with lap joints. 

From (397) we have : 



p = 368,000 



0.25 



V: 



0.25 

=5" X 23 



= 303 lbs. 



at whicli pressure the flue actually collapsed. It is evident that should the 
thickness of the flue be only slightly reduced by corrosion, etc., an explo- 
sion might readily follow. 

A method of increasing the safety without using a greater 
thickness of metal in the walls of the flue, is to reinforce it by 
stiffening rings, thus practically reducing the length /, as noted 
by Fairbairn. 





Fig. 1 1 19. 

Two forms of stiffening rings are shown in Fig. 1119, a being 
Adamson's and b, Hick's. The first form is the more difficult 




Fig. II 20. 



Fairbairn deduced from his experiments for the collapsing 
pressure of such flues : 



p' = 806,300 



r! 2-19 



(395) 



in which p' is the pressure in pounds per square inch, D and i 
are in inches, and I is the length of the flue in feet. 

If the dimensions are given iu millimetres aud/i' is the pres- 
sure in kilogrammes per square millimetres, this becomes : 



100 p' : 



367,973 



(I ■'■■9 

Tn 



(396) 



Fairbairn's experiments have been discussed more recently, 
■with a view of deducing a formula which should be more con- 
venient to use.t The results of Dr. Wehage in connection with 
later experiments, J give the following formula : 



r- 



368,000 
490,000 



J ^ 






(397) 



in which the upper coefficient is to be used for flues made with 
lap joints, riveted ; and the lower coefficient for flues in which 
the joints are made with flap plates riveted on. 

This formula gives results approximating verj' closely to 
Fairbairn's most important experiments. It is best used by 
selecting the desired dimensions for D, I and '5 and then by sub- 



* The question has been raised as to whether it is not best to stay only 
one boiler head to the shell and then tie the other head to the first by means 
ot a number of parallel anchor bolts, thus closi \\^ one end of the shell in a 
manner similar to the cylinder of a hydraulic press, and relieving the shell 
of any stress due to the pressure on the heads, and permitting the use of 
packing to make the joint tight. The author recollects such a construction 
having been used in a portable engine and boiler but without knowledge of 
any further attempts of the sort. 

t See Grashof, Zeitschr. D. Ing. 1S59, P. 234. Vol. III. ; also Love, (-ivilin- 
genieur, 1861, p. 238, Vol. VII,, discussed by tlie author for the HUtte Society 
in the Berliner Verhandlungen, 1870, p. 115. 

t See Engineer, Vol. 51, 18S1, p. 426, also Dingler's Journal, Vol. 242, 1881, 
p. 236. 



of construction, but possesses the advantage of removing the 
rivet heads entirely from the action of the fire. This form of 
joint and stiffening plate is also frequentlj' used in other parts 
of boilers for the sole purpose of avoiding the action of the fire 
on the heads of the rivets. 

The use of corrugated iron for boiler flues enables great 
strength against collapsing to be obtained. Fig. 11 20 shows a 
boiler with corrugated flue, the lengths being welded together. 
This boiler is made by Schulz, Knaudt & Co., of Essen, and is 
86.6 inches in diameter (2.2 metre). Notwithstanding the con- 
structive difficulties the use of the corrugated flues is constantly 
increasing. In England corrugated flues are made by the in- 
ventor, Sampson Fox & Co., of Leeds. The depth of corruga- 
tions is usually about 4 inches. 

Corrugated fire boxes have been used in locomotive boilers. 
Fig. 1121, showing Kaselowsky's fire box. In this form the 




Fig. 1 121. 

stay bars to support the crown sheet, and the stay bolts at the 
sides are entirely omitted. The cross section shows the method 
of supporting the boiler by a cross beam below the grate bars. 
The corrugated flue is attached to the boiler by a riveted joint, 
either by flanging as in Fig. 1121, or by the use of angle iron, 
as Fig. 1120. 

Tubes of small diameter are treated practically as single hol- 
low rivets, the ends being inserted into holes in the tube sheets 



270 



THE CONSTRUCTOR. 



and expanded by an expanding tool, the ends being riveted over> 
as shown in Fig. 1 122 a. 



Fig. T122. 

In many establishments, as for example, the Esslingen loco- 
motive works, the tubes are fitted with hard copper ferules 
which stand the expanding and riveting better than tubes of 
steel or iron. The form of tube shown in Fig. 1122^, is rein- 
forced at the ends, and one end made conical, thus enabling 
old tubes to be more readily removed and replaced. This con- 
struction is used by Pauksch & Freund_, of Landsberg, in Ger- 
many, and by various French builders since 1867. 

I 362. 
Future Possibilities in Steam Boiler Construction. 

The discussion of the preceding sections has necessarily been 
limited to a few constructive details, since a complete treatment 
of such an extensive subject requires a special treatise. It is 
proposed here to give only 'a broad general view of the subject 
of boiler construc"tion in its present and prospective condition. 

The descriptions in the preceding sections and in the previous 
chapter on riveting show that the art of boiler construction has 
made little or no advance during the past twenty or thirty years, 
although there is reason to believe that there is ample room for 
improvement, especially in the matter of greater economy of 
fuel. In the author's opinion there are four points in construc- 
tion which deserve the closest attention and to which efforts at 
improvement should be directed, while in other directions also 
serious wastes of force appear. 

1. Expenditure of Material. — As already shown in ? 359, the 
expenditure of material is considerably greater in the present 
forms of steam boilers than if the spherical form were more 
generally used. It is questionable to what extent the spherical 
form may be made practicable, but the possibilities in this direc- 
tion have not been exhausted, at least for certain purposes, for 
example, for boilers used solely for heating purposes. The 
spherical vacuum pans only serve as reminders that this oldest 
form of boiler (i. e., that used with Newcomen's engine), is no 
longer used ; but it may be only a question of the increase in 
the capacity of the flanging press ; or, in other words, of the 
increased command over the working of iron and steel, when 
the spherical form shall again be used. 

Another point in the question of material, is the subject of 
riveting. One of the greatest sources of weakness in steam 
boilers is the reduction in strength due to the presence of riveted 
seams. Even if the very best material obtainable is used for the 
rivets, the reduction in strength for single riveting is about 40 
per cent, and for double riveting, 25 per cent.* This weaken- 
ing is unimportant so far as the circumferential seams of cylin- 
drical shells are considered, but is well worthy of consideration 
in connection with the longitudinal seams, especially since it 
concerns the largest and heaviest part of the boiler, i. e., the 
main shell. It is for this reason that attempts have been made 
to weld the longitudinal seams. 

The meagre results which have been obtained for welded shells 
subjected to internal pressure, as compared with welded flues 
for external pressure, may be seen from the case shown in Fig. 
1116. The welded seam is there reinforced by a riveted flap, 
thus reducing the strength practically to that of an unwelded 
seam. Experimental results with welded joints in the testing 
machine, justify this distrust of welded seams, and do not war- 
rant the idea that the weld is equal to the full strength of the 
plate. 

This leads to the remark that the coming boiler shell must be 
without longitudinal seams of any kind, either riveted or welded^ 
Heating flues for external pressure are already made seamless, 
and the Mannesmann process produces seamless tubes adapted 
for internal pressure, and of a grade of material far superior to 
that heretofore used, as experimental researches have demon- 
strated. If this process can be so extended as to be made avail- 
able for boiler shells, an economy of at least one-third of the 
material can be obtained. 

2. Combustion. — The subject of economy of combustion of 
the fuel is even more important than that of material. In the 



general description given in the preceding sections it will be 
seen that the present methods of firing are all based upon the 
principle of exposing portions of the boiler to the direct action 
of the fire and of conducting the products of combustion into 
contact with various portions of the boiler, arranged to act as 
heating surface. This means that in nearly all cases boilers are 
independently fired. For a long time the advantages of this 
system have been doubted. It is manifestly impossible for a 
complete combustion of the gases to be effected when they are 
almost immediately brovight into contact with surfaces which 
have a temperature of 1200 to iSoo degrees lower than the flame. 
The production of smoke and soot, that is, of unconsumed fuel, 
is the necessary result of these conditions, and hence a great re- 
duction in efficiency. This subject has been actively worked 
over, and an almost endless variety of furnaces and systems 
has been proposed. The true method of solving the problem 
appears to have been first discovered b}' Frederick Siemens 
(Dresden), and for a number of years he has been engaged in 
developing the practical applications of his researches, t 

The previous methods of firing were based upon the idea of 
bringing the flame into direct contact with the surface to be 
heated, but since about 1879 the method of construction, espe- 
cially in glass furnaces, open hearth steel furnaces, smelting 
furnaces, etc., has been to utilize the radiant heat from the 
arched roof of the furnace, and to economize the heat of the 
escaping gases in the regenerator. An ecouomj' in the use of 
the heat of as much as So to 90 per cent, has resulted. This has 
been followed by a still more marked separation between the 
two principal periods of combustion, and by the application to 
steam generators where such a high economy cannot be expected, 
although a saving of about 25 per cent, has been shown in actual 
practice. I 

It is therefore strongly recommended to use such furnace con- 
structions as shall not bring the direct flame of the fire in con- 
tact with the heating surface of the boiler, but to use radiating 
surfaces and also to conduct the highly heated but full}- burned 
gases through the flues, both of which can be accomplished in 
various ways.? 

The application of the principle to stationary boilers is not 
difficult, and experiments have shown that it may also be suc- 
cessfully applied both to marine and locomotive boilers. In all 
cases it has lieen demonstrated that the fuel should be burned in 
a combustion chamber lined with refractory material, and the 
discharge of the heated gases retarded by a fire brick bridge or 
screen before coming in contact with the boiler. It will be seen 
from the preceding, that by using the Siemens' method instead 
of the older method of burning the fuel directly in the boiler, 
an economy of about 25 per cent, can be obtained, and this fact 
should always be kept in mind in future designs. 

3. Heating Surface. — The third point concerns not so much 
a variation in construction, as it does the lack of knowledge of 
the fundamental principles, this suljject having been much less 
fully investigated than other portions. Recent investigations 
show conclusively that the axiom that the heating surface is a 
magnitude proportional to the desired efficieucy of the boiler, 
cannot be sustained. It is evident that there must be a very 
considerable difference in the heating value of portions of the 
surface which are at greatly different distances from the fire. A 
very high temperature of the gases at the beginning, and a 
comparatively low temperature near the end, must mean a rapid 
formation of steam near the fire and a weak production over 



*When the rivets are made of no better material than the plates, the re- 
duction for single riveting is about 53 percent., and for double riveting 
about 41 per cent. Triple riveting, as shown in Fig. 155, is too expensive to 
come into general use. 



fThe following list will serve for those who aesire to refer to the original 
and fundamental publications upon this subject : — Friedrich Siemens, Heiz- 
ver fohren niit freier Flamment faltung, Berlin, Springer, 1882; Siemens' 
Regenerativofen, Dresden, Ramming. 1S54 ; Vortrag von Friedrich Siemens 
Uber Ofeubetrieb mit ausschliesslicher Benutzung der strahlenden Warme 
der Flamme, Gesundheitsingenieur, :8S4 ; Vortrag von demselben iiber ein 
neues Verbrennungs-und Heiz-systeni, Busch, Journ. f. Gasbeleuchtung, 
etc., 1885; Vortrag von demselben in der Ges. Isis in Dresden iiber die Dis- 
sociation der Verbrennungsprodukte. Dresden, Blochmann, 1SS6; Vortrag 
von demselben im Siichs. lug. u. .\rchit Verein iiber die Verhiitung des 
Schornsteinrauches, Civ. Ing. Bd., 32, Heft 5, 1886; Vortrag vom demselben 
ini BezVer. D. Ing in Leipzig am 8 Dez. 1886 Uber den Verbrennungsprozess, 
2 Aufl , Berlin, Springer, 1S87; Vortrag von demselben, gehalten in Ham- 
burg im Ver. D. Gas-und Wasser fachmanner iiber Regenerativ — Gasbrenner, 
etc., Dresden, Ramming, 1887; Ueber die Vortheile der Anwendung hocher- 
hitzter Luft fiir die Verbrennung, etc, 2 Aufl., Berlin, Springer, 1887, 

X For example, a test by K. H, Kiihne & Co., of Dresden Lobtau on Feb, 
16, 1884, showed a gain of 26 per cent, due to the substitution of a Siemens 
furnace for one of the usual kind ; the conditions of draft and cleanness of 
flues being alike in both cases. 

g Two methods have been described by Dr. Siemens, both of which have 
been applied by him to flue boilers. In the first, the combustion of the fuel 
takes place upon a grate in a combustiou chamber which is directly over 
the grate. A bridge wall of fire brick is placed about half the length of the 
grate further back, and beyond this are two ring shaped screens of fire brick, 
which are so placed as to direct the products of combustion toward the axis 
of the boiler flue ; after passing through the flue the gases return about the 
outside of the shell and are then sufficiently cooled to be permitted to pass 
over the portions of the shell unprotected by water on the way to the chim- 
ney. In the second method the fuel is burned to gas in a gas producer 
separately constructed from the boiler, and the gas mixed with heated air 
and thus delivered to the boiler flue, where it follows the same course as in 
the first case. 



THE CONSTRUCTOR. 



271 



distant portions of the surface. It lias been shoT\-n that in some 
instances the heating surface of one and the same boiler ma}- be 
reduced one-half without caxising any reduction in the steam 
production. The usual method of proportioning the heating 
surface in all kinds of boilers appears to be based upon previous 
results with similar forms, and hence is often one-sided and 
unsuited for sj-stematic investigation. A new departure in the 
discussion of this important subject has been made by the chief 
director and engineer of the Swedish railway's, Mr. F. Almgren. 
He has made the subject of the proportioning of heating sur- 
face the object of a series of experiments extending over a 
number of j'ears, and has placed the matter upon a much higher 
plane of investigation than heretofore. The practical results 
are of much importance, and in advance of the publication of 
the whole the following general discussion has kindly been 
placed in the author's hands by Mr. Almgren, and is here given 
in his own words.* 



PRACTICAL RESEARCHES UPON LOCOMOTIVE BOJLERS WITH 
SMALL TUBES. 

BY F. AI<MGREN. 

"According to the investigations of Geoffro}-, as given by 
Couche,t the amount of steam produced by tubular heating sur- 
face depends upon the volume of heated gases passing through 
the tubes per hour. The heating surface under experiment con- 
sisted of portions 0.9 metre long of tubes, the total length of 
which was 3.6 metres long each. 

" I have found that the volume of gases may be considered as 
a function of the length / of the tubes, the latter being con- 
sidered as a variable, according to the following general e.xpres- 
sion ; in which i is the number of tubes, and L the number of 
heat units given off by each tube per second. 



a 

TzT 



(398) 



" In this formula a and b are constants which depend upon 
the mean temperature Te of the gases, upon the temperature (5 
of the water, and upon the weight / G of the gases passing 
through the tubes per second. 

" As the result of a series of experiments I have found these 
constants as follows : 



« = 0-357 iG [Te — 6) \ 
b = 7.15 G"°-=i7 J 



(399) 



in which G is the mean weight of gases or products of combus- 
tion for one tube. For the number of heat units L given off by 
a single tube of a set, the following expression is given : 



L ■■ 



, 0.357 G { Te- 6) 
I + -^^^ Co.^i? 



(400) 



"In order to show the utility of these formula, a table is 
here given of the results of twenty-one experiments upon a 
locomotive boiler, the walls of the fire box having been made 
non-conducting by means of brick-work. A second table is also 
given to show the great advantages resulting from these experi- 
ments. The quantities given in the table are as follows ; 

i =- the number of tubes. 
G = the weight iu kilogrammes of the pi-oducts of com- 
bustion passing through each tube per second. 
Tr — (5 = the difference between the temperature of the smoke 
and the water, the former being measured in the 
smoke box. 
Te — c! = the difference between the mean temperature of the 



gases in the tubes and the water = T, — rf + 



L 



Le-- 
Lb-- 



: the mean value of L determined b 

: the value of L determined by formula (400). 



0.24 
experiment. 



G. 



* It has been thought best to leave the formula; and tables in the metric 
system, and temperatures in the Centigrade thermometer, also keeping the 
French thermal units, and thus retaining the discussion in Mr. Almgren's 
own figures, as the principles are equally well shown, and the unity of this 
preliminary presentation thereby retained. — Trans. 

t Voie. materiel et exploitation des Cbemias de fer, Tome III. 



TABLE I. 

Locomotive boiler: pressure 4 atmospheres, tubes of brass, 2.934 metres 
long, 42 mm. diameter, somewhat scaled. 



No. 


i 


G 


TV — <! 


Te-& 


Le 

1. 184 


Li 


I ^ 






0.00713 


210° C 


901° (T 


1.248 


2 






0.00601 


1S5 " 


916 " 


1-035 


1.090 


3 




no 


0.00733 


222 " 


969 " 


1.304 


I -370 


4 






0.00827 


230 " 


1009 " 


I-53I 


I-570 


5 . 






0.00900 


23s " 


1000 " 


1.648 


1.700 


61 




0.01795 


275 " 


1067 " 


3-360 


3-330 


7 






0.01S71 


285 " 


I09I " 


3.600 


3-520 


S 






0.01S32 


278 " 


III5 " 


3.660 


3-530 


9 \ 


55 


0.01479 


290 " 


I42I? 


4.000? 


3-750 


10 




0.01514 


240 " 


1221° C 


3-5IO 


3.290 


II 




0.01303 


255 " 


I312 " 


3-300 


3.080 


12 J 




o.oiogi 


235 " 


1328 " 


2.860 


2.700 


13 1 
14 




0.00466 


90 " 


682 " 


0.650 


Q.646 


88 


0.0044S 


95 " 


724 " 


0.670 


0.660 


15 
16 J 


0.00405 


95 " 


781 " 


0.660 


0.652 




0.00360 


95 " 


709 " 


0.530 


0.530 


17 " 






0.00586 


75 " 


462 " 


0.542 


0.534 


18 






0.00529 


70 " 


368 " 


0.376 


0.388 


19 


' 


no 


0.00640 


83 " 


466 " 


0.591 


0.586 


20 






0.00715 


95 " 


522 " 


0.734 


0.721 


21 






0.0066S 


90 " 


529 " 


0.695 


0.686 



"Remarks. — Between each set of experiments the boiler -n'as 
blown off and both boiler and tubes cleaned. The no tubes of 
the fourth set were only partially the same as those of the first 
set. In the ninth experiment one of the cast iron plugs which 
were used to close the tubes not in use was melted out. 

" The correspondence between the experimental value Le and 
the calculated value / * is ver_v striking. A formula for special 
practical cases has also been deduced, being adapted for the 
special number of tubes as given in the preceding table, and 
without the variation in G and Te which occur in single experi- 
mental cases. 

" Equation (400) shows that for a given length / for the tubes, 
the production of steam is nearly proportional to the weight of 
gases flowing through them, and that it also increases nearly in 
direct proportion to the quantit}' of heat 0.24 G (Te — 6). 
This indicates that for a constant blast opening, the amount of 
steam produced by the heating surface of the tubes will almost 
exactly equal the amount of steam passing through the blast 
nozzle, that is the amount of steam used bj' the engine. If it is 
also remembered that an increase of draft also increases the 
temperature of combustion, k will be seen that the tubes and 
the blast nozzle of a locomotive boiler bear a most intimate 
relation to each other, and that great and sudden variations in 
the production of steam occur almost hourly. 

" Now the researches of Geoffroy show that the walls of the fire 
box have a much less favorable action. In this portion of the 
heating surface the production of steam responds much more 
slowly to variations in the draft. The larger the fire box, the 
more marked is its action in this respect, and consequently the 
less effective will be the blast. Equation (400) shows that for a 
given tube length, the production of steam of each tube in- 
creases with the increase of the draft, and hence the number of 
tubes and consequently the weight of the boiler may be kept at 
a determinate minimum, which depends upon the permissible 
force of blast and limit of size of grate. The formula also shows 
that with a strong draft and high temperature even the latter 
portion of long tubes is of excellent .steaming value. 

" Since also a given amount of tube heating surface is lighter 
and cheaper than the same amount of fire box surface, and since 
by the reduction of the latter the products of combustion will 
be cooled less and so enter the tubes at a higher tempera- 
ture, it will readily be seen that a material advantage can be 
gained by removing that portion of the fire box surface which 
is of the least value (that is, the side walls), and adding an 
equivalent proportion by lengthening the tubes. As an exam- 
ple may be cited the case of a locomotive boiler wdth 125 tubes, 
3 metres long and 45 millimeters inside diameter iu which a re- 
duction of 7 square metres of fire box surface was made up by 
an increase in length of tubes which gave 14 square metres of 
surface, the force of draft being 40 millimetres water pressure. 
This change removed the expensive stayed fire box walls, which 
were replaced by a fire brick lining, and the reduction in weight 
and cost amounted to about 700 kilogrammes and 1500 marks. 



272 



THE CONSTRUCTOR. 



' ' The latest boilers for the Swedish State Railways have been 
constructed with the preceding principles in view as shown in 
Fig. 1123. The fire brick lining of the fire box is shown at a, a, 




Fig. 1 123. 

•while at b, b, are openings for the admission of air, which can 
be closed lay sliding dampers c. A 5'ear's experience with this 
construction has given satisfaction, as the following table shows. 
It will be seen that the new form of boiler produced the same 
amount of steam per unit of heating surface as the old form, the 
force of the draft and the temperature in the smoke box being 
nearlv the same in both instances. 

"The external length of this fire box is 1.4S5 metres, and the 
internal width is i metre. The diameter of the shell is 1. 103 
metres, with 144 tubes, 45 millimetres inside diameter, and the 
fire brick lining is 74 millimetres thick. 

"TABLE II. 

A. Dimensions. 





Tubes. 


Heating Surface. 


Boiler. 


Length. 


Diame- 
ter. 


No. 


Tubes. 


Fire Box. 


Ratio. 


OldStyle.3.111 m. 
New " ,3.305 m. 


46 mm. 
46 mm. 


184 
102 


77.28 sq.m, 
50.83 " " 


7.S2sq. m. 
2.19 " " 


9-9 
23.1 



B Performance. 



Evaporation 


Draft Pressure in millimetres 
of water. ' 


Temperature in Smoke Box. 


per hour 


Old Style 


New style 


Old Style. 


New Style. 


24 •*.?■■ 
30 " 

37-45 " 
55 " 


20 mm. 
30 " 
40-50 " 
80 " 


24 mm. 

35 " 

50-60 ' ' 

90 " 


310° C 
340° 
410° 
470° 


315°^ 
340° 
395° 
470° 



" A patent has been applied for by Herm. Von Storckenfeldt 
for the construction shown in Fig. 1123, and made from my cal- 
culations and directions." 

The preceding brief description shows the nature and import- 
ance of Almgren's researches and appears to form a starting 
point for a change in methods of locomotive boiler construction. 
Further investigations may develop a theoretical foundation 
for this empirical formula. Especially interesting is the con- 
formity of Almgren's observations with the above described 
results of Siemen's. We also see the previous remarks upon 
the subject of economy of material confirmed in the advantages 
resulting from the replacing of flat stayed and riveted surfaces 
by c}'lindrical welded tubes. A corresponding gain would be 
attained were it possible to produce a shell free from riveted 
seams. 



4. Artificial Draft. — The use of forced draft has been com- 
mon for many years in locomotives and portable engines, and 
by this means a much greater quantity of steam produced from 
a unit of heating surface than with natural draft. More recent- 
ly forced draft has been applied to marine boilers, the blast 
generally being produced by fan blowers. Especially has this 
been necessary in the case of torpedo boats, in which the high- 
est speed is demanded. By the use of multiple expansion en- 
gines operated by greatly increased steam pressure speeds of 18 
to 20 knots are attained without an excessive increase in the con- 
sumption of fuel. This, however, involves a much greater in- 
crease in the steaming capacity of the boilers in proportion to 
their weight, and this result is accomplished by the use of arti- 
ficial draft. 

This has been discussed very completely in a paper presented 
before the Royal United Service Institution, by Naval Engineer 
H. J. Oram, upon the subject of the motive power of modern 
war ships. The large boilers of the English war ships ' ' Blen- 
heim " and " Blake " are 15 feet in diameter and 18 feet long, 
with four furnaces at each end These are worked witk closed 
ash pit, and an air pressure of two inches of water, and at a 
steam pressure of 150 to 165 pounds, the engines indicating 
3,350 horse power. The air pressure of two inches is ample for 
the desired rate of combustion, and bj^ reference to the prece- 
ding table it will be seen that it is no greater than has long 
been common in locomotive practice. The combustion is more 
complete under this pressure than with natural draft, being 
more uniform and producing less smoke. It may be remarked 
that the efficiency of the boilers maj' also be increased by pro- 
per heating of the feed water and by use of the double distilling 
apparatus. The use of forced draft also makes it practicable to 
cool and ventilate the stokeholds. 

The latest examples of construction, of American design, are 
made to work at pressures as high as 250 pounds per square 
inch, with boiler shells 16 to 17 feet in diameter. Mr. Oram 
considers that there is a limit to increase in this respect due to 
the increase in weight beyond practical limits, both of the boil- 
ers and of the engines. 

It is worthy of note that in the recent express steamers of the 
French " SocietiJ des Messageries Maritimes " the use of shell 
boilers has been abandoned, and sectional boilers of the Belle- 
ville type introduced. The increase in speed also appears to 
have its limits, but the advantages of forced draft, however, as 
regards the reduction in size and weight of the boiler, should at 
least lead to its introduction in the future for stationary prac- 
tice. 

Taking into consideration all the points of the preceding dis- 
cussion, it appears that an application of them to practical boil- 
er construction should result in an economy both of construc- 
tion and of operation of 25 to 33 per cent, with entire safet". 



I 363- 
Reservoirs for Air and G.\s. 

In the use of compressed air now so general in mining and 
tunneling operations, C3'lindrical reservoirs similar to steam 
boilers are used In tunnel construction, portable reservoirs are 
sometimes foinul mounted upon tram locomotives, the engines 
of which are operated by the compressed air instead of steam. 
Compressed air locomotives have only been used to a small ex- 
tent, however, for general tram service. The so-called pneu- 
matic method of sinking shafts and construction piers involves 
the use of air reservoirs. In this case the air reserv'oir is the 
caisson within which the work is carried on, the water being 
kept out by the air pressure, and the workmen entering and 
leaving b}' an air lock chamber with a double s\-stem of doors. 

In the case of power transmission in cities Iw means of com- 
pressed air, the entire sj'stem of piping is included in the reser- 
voir capacity. Negative reservoirs for mingled air and steam 
are found in the case of condensers for steam engines. These 
are usually made of cast iron and are from one to two times thfe 
capacity of the steam cylinder. The regular removal of the 
contents by the air pnmp'at each stroke of the engine renders a 
larger capacity unnecessary. In some cases the flow of spring 
has been increased by fitting a tight cover over the well above 
the water level when the exhaustion of the air causes an in- 
creased flow from the underground sources. The vacuum sys- 
tem of power distribution, as used in Paris and London, in- 
volves the use of negative reservoirs similar to C3-lindrical 
boilers. An important application of vacuum for air and vapor 
of water is found in the vacuum pans used in sugar refineries. 
These pans are made in the spherical form, already referred to 
as most economical of material, the motive in this instance 
being the high price of copper, of which they are constructed. 
Gas holders for illuminating gas are reservoirs intended only 
for very low pressures, the strength of the walls being most 



THE CONSTRUCTOR. 



273 



important in the matter of tightness against leakage. These 
holders are composed of two principal parts, the holder proper, 
or so-called " bell," often made telescopic, and the tank or res- 
ervoir filled with water which acts as a liquid packing ; the bell 
in this case acts as a piston (compare Fig. 948). Similar reser- 
voirs are used in laboratories and chemical works for many 
kinds of gases. For very large gas holders, in which the inter- 
nal pressure of the gas is insufficient to sustain the weight, the 
roof of the holder must be strengthened by internal trussing. 
Until now the gas holder has had no definite place in construc- 
tion, but it will be seen from what has already been said, that 
it, together with various other kinds of reservoirs, belong pro- 
perly to machine construction, not only because of their char- 
acter but also because of their intimate connection with the 
entire subject of mechanical engineering. 

? 364. 
Other Forms op Storage Reservoirs. 

The construction of reservoirs for water has been a most im- 
portant subject from the earliest times down to the present, 
many of these being of great extent, although, as has already 
been said, these have until now been considered rather as be- 
longing to the domain of building construction than to machine 
construction. To these must also be added the subterranean 
reservoirs in mines, from the small pump to those of large 
extent and capacity. Other examples are found in the negative 
reservoirs which exist in low-lying tracts of land, such as are 
found in Northern Germany and Holland, intersected by canals. 
A notable example in Holland is the valley formed by the 
drainage of the Harlem Lake, the water having been pumped 
by steam engines out to the level of the sea and the latter kept 
out by d3'kes. 

Reservoirs for agricultural purposes are often formed by sys- 
tems of canals, as in Lombardy and ia the south of France, 
where this important svibject of irrigation has proved of the 
greatest benefit to the country. The nature of such S3'stems, 
considered as reservoirs, is more apparent when the magnitude 
of the work involves the construction of artificial lakes for 
water storage. Ancient examples of such storage reservoirs 
are found in Lake Moeris, of ancient Egypt, and Lake Nitocris, 
of Babylon, as well as the existing Lake Maineri, in Ceylon, 
and many others. The mechanical nature of such constructions 
is more apparent when the reservoir is made by building a dam 
across a gorge or valley, with weirs to permit the periodical 
release of the water, the analogy to ratchet action being quite 
clear. 

Finally, another natural form of stored power may be men- 
tioned, one which has not to the writer's knowledge been con- 
sidered in this light before, j'et which possesses the greatest 
significance in the climatic economy of nature. This is the 
glacier. The vapor of water, raised from the level of the sea 
by the heat of the sun, collects in the form of snow about the 
highest mountain peaks. In the upper valley's the snow packs 
together, aud under gradual pressure forms the glacier ice, and 
slowly the glacier flows down into the lower and warmer val- 
leys and melts away. The mass of ice, consisting of hundreds 
of millions of cubic feet, forms a reservoir of stored power, 
flowing in an irresistible stream of almost uniform strength 
from the highest snow field to the lower valley. All the actions 
involved are of a physical and mechanical nature. Taken as a 
whole the glacier forms a reservoir system of the fifth order : 
evaporation of the water from the sea by the heat of the suu, 
transformation of vapor into snow, fusion of the snow into a 
mass, conversion by pressure into glacier ice, and melting of 
the ice partly by the friction on its iSed and partly by the heat 
of the sun. 



CHAPTER XXVI. 



RATCHETS FOR PRESSURE ORGANS, OR VALVES. 
?365- 

The Two Divisions of Vai^ves. 

The application of the ratchet principle to pressure organs, 
that is, the periodical interruption of its motion, closely resem- 
bles the same principle applied to constructions formed of rigid 
elements ; the principal difference being that the pressure organ 
is very easily separable into small portions. It might also be 
remarked that tlie pressure organ is alwaj's confined in a con- 
ductor of some kind, but this feature also belongs to some forms 
of rigid constructions, such as bearings, guides and the like. 

Ratchets for pressure organs may be divided into two princi- 
pal classes, namely, those intended to check the motion in only 
one direction, and those which check in both directions. The 
name given to ratchets for pressure organs is valves.* 



The difference between the two classes is shown in Fig. 11 24. 
In the form shown at a, the pressure organ is checked by the 
flap valve b, from moving in the direction of the arrow at /, 





Fig. 1 124. 

but not against motion in the direction of the arrow at //. In 
the form shown at b, the flow is checked in both directions. 
There is here a close analogy to the two kinds of rigid ratchets, 
as will be seen in Fig. 1125, which is here reproduced from 
? 235. The valve b, in Fig. 1124 <z, corresponds to the pawl in a 

Running Ratchet, 

and the valve in case 5, to a 

Standing Ratchet 

for the pressure organ a. The difi^erence in construction will 
also be seen to depend upon the fact that in form a, the valve 
lifts from its seat during the passage of the pressure organ, 
while in form b, the valve slides upon the seat. This permits 





Fig. 1125. 

another classification into: 

a. Lift valves ; 
(5, Slide valves. 

The variety of forms in which valves are constructed is fully 
equal to that of rachets for rigid elements, as shown in Chapter 
XVIII., and there is a close analogy existing between the two 
groups, with one important exception, nam el j', that the form 
of rigid rachet which has a tension pawl, has no counterpart 
among the valves. This exception naturally follows from the 
fact that the member to be checked is always subject only to 
compression. 

There is also an analogy between the numerous forms of 
valves and the two classes of toothed and friction rachets, as 
has already been mentioned in | 319, valves which have but a 
slight opening, acting like friction rachets (compare § 340), and 
those with full opening and entire closing like toothed rachets. 
This circumstance, however, reduces the number of sub-divi- 
sions into which valves may be classified, so that the principal 
basis upon which a classification is made depends upon the 
character of the motion of the valve, and thence upon the 
necessary variation in form. This basis of classification has not 
been used in the case of rigid rachets, the divisions there 
having been made upon the more practical idea of the variation 
in form only. We have in rigid ratchets the two forms of 
pawls, one of which moves about an axis 3 within a finite dis- 
tance, as in Fig. 11 24; and the other in which the axis is re- 
moved to an infinite distance. In the first case, ever}' point of 
the pawl (or valve) moves in circular arc about the axis, while 
in the second, all points move in straight lines and equally far. 
In rigid systems these correspond to link pawls and bolt pawls. 



*The author calls attention to the derivation of the German word " ven- 
tile,'' from the medieeval name for valves used for checking wind in church 
organs. The English word *' valve'' from the I^atin "vulva," meaning: 
hinged doors, is therefore broader and more general — Trans. 



274 



THE CONSTRUCTOR. 



In addition to the circular and rectilinear motion of valves, 
there is a third variety possible, although but little used in 
practice, viz, : those having a spiral motion. We therefore 
have three sub-divisions of the two main classes of valves, 
according as the movement is circular, rectilinear, or spiral. 

Lift valves may be 

1. Hinged oi Flap Valves. 

2. Disk, Cone, or Ball Valves. 
3- Spiral Lift Valves. 

Slide valves may be 

1. Rotar_v Valves or Cocks. 

2. Rectilinear Slide Valves. 

3. Spiral Moving Slide Valves. 

Although this sub-classification is not exhaustive, yet it gives 
a convenient and practical arrangement, the few special forms 
being placed m the group they most nearly resemble. 

.-1. LIFT VALVES. 

I 366. 

HINGED OR FLAP VALVES. 

Flap valves are most generally applicable to piston pumps, 
which, as we have already seen, form fluid escapements, see 
^319. Their tightness is often attained by the use of some 
elastic material, such as leather, rubber, etc., but very generally 
the joint is made between metallic surfaces, especially' when no 
small hai'd particles are likely to be found in the passing fluid. 
It is always difficult to keep the loss due to shock within small 
limits, this loss being especially marked with flap valves, and 
indeed in all liquid ratchet systems the loss from this cause is 
by 110 means unimportant. 



The width of bearing s of the valve on its seat is given \>y the 
following formula, in which D is the clear opening through the 
valve. 




j = \/i? + 0.16' 



(401) 



For round valves D is the diameter of the opening ; for rec- 
tangular openings it is taken as the smaller side of the rec- 
tangle. 

The blow with which a valve strikes the seat increases in force 
with the amoinit of lift (compare \ 368), and as the lift depends 
upon the actual size of the valve, this objectionable feature is 
reduced bj' using several valves of smaller size instead of a 
.''i.ngle large one. 



Fig, 1 1 26, 




A flap valve with metal seat, which is so constructed as to 
offer as little obstruction as possible to the flow of liquid, is 
shown in Fig. 1126*. This is tapped out for the standard pipe 
thread system described in J 342, the cap gives access to the 
vafve, the screw plug limits the amount of lift, and a flexible 
connection between the disk and the hinge enables the former 
to obtain a fair bearing on its seat. The freedom from shock 
would be somewhat less if the bottom of the case conformed to 
the shape indicated by the dotted lines. 



<t5.-%8- 




Fig. 1 127, 

Another form of straight-waj' flap valve is shown in Fig. 1127, 
Both valve and seat are made of bronze, the seat being secured 
in place by two wrought iron kej'S. The case is closed by a lid 
shown removed in the illustration. The axis of the valve is 
made to permit a slight degree of lateral play in order to permit 
the best bearing on the .seat to be obtained. Valves of this .sort 
are used on air pumps for steam engines and for vacuum pans. 



A double flap valve and valve chamber designed foramine shaft 
pump, is shown in Fig. ii2S(r. The flaps are formed of pieces 
of leather between plates of iron, secured either bj' screws or 
hy rivets. The door by which access is obtained is curved to 
the shape of the valve chamber in order to avoid excessive dead 
.space, and so reduce the shock, and is supported upon hinges. 
The stops are so placed that the valves open to an angle of 
60°. 

Another design for a double flap valve is shown in Fig. 1128^, 
this also being for a shaft pump.f In this instance the valves 
are formed of three thicknesses of leather. At c is shown a 
quadruple valve. The proportions given are all based upon the 
imit i, as given by formula (401). 

i Fig. 1129 shows a circular valve of 

rubber, this form being much used 
for air pumps for steam engines. The 
.1 valve lifts approximately in a circular 
'* path, forming a cup, the limit of 
which is the shape of the guard. On 
account of the flexibility of the rub- 
ber, the bearing of the seat is rein- 
forced b}- a grating, and the rubber is 
from Yi to \Vi inch in thickne.ss. 
These values are now made also of 
vulcanized fibre, in which case the thickness ueed be only about 
one-third that of rubber disks of the same diameter, J 

Quite similar in principle to the above di.sk valve, is the 
leather rolling valve, Fig, 1130(7, u.sed for water wheel gates, the 
principal difference Ijeing that the bending of the valve takes 
place at the edge of the valve, as shown in the illustration. 




Fig. 1129. 



* Pratt's Straight-way Check Valve, 



fSee Riebler, Tndikator versuche an Purapeu und Wasserhaltungs ma- 
schMieii, p, 34, IMunich. 18S1, 

X Made by tlie Vulcanized Fibre Company of New York. 



THE CONSTRUCTOR. 



275 



The same principle is ingeniously used in the hanging weir 
of Camere,*Fig. \i2,ob. The valve consists of a series of strips 
of wood, each really forming a separate valve, these being con- 
nected and operated by chain links of bronze as indicated in 
the sketch. 

b. 




Fig. 1 130. 

An excellent installation is seen at the sluice gates at Geneva 
(Passerelle de la machine), where forty such gates are used to 
dam the right arm of the Khoue. The gates are rolled up by 
the chains shown, these being connected to suitable windlasses. 
When a whole section is to be thrown entirely open the support- 
ing posts are also tipped back into the horizontal position, 
these being jointed at the bottom as shown, and this operation 
being effected by another chain gearing. Each gate is 3 ft. 8 
inches wide ; the sets of connecting links are 27 Ji inches apart, 
the number of strips is 39, each being about-3 inches wide, the 
uppermost being 24-g inches thick, and bottom one 3}^ inches. 

The weir sj'stem at Geneva, of which the above forms only a 
small portion of the entire work, was completed in 18S9, as an 
intercantonal system to control the level of the lake of Geneva 
and maintain it between the limits of 1.30 and 1.90 metres (4 ft. 
3^ in. and 6 ft. 2)^ in.) of that of the Rhone. During the year 
1888, when the system was not entirely completed, the differ- 
ence fell to i .95 metres (6 ft. 434" in.) in the drought of June 
of that year. Between October and May the entire series of 
gates was kept closed. 

^367. 
Round Sei,f- acting Valves. 

Lift valves for small openings are frequently made of con- 
ical or spherical form, and in Fig. 1131 two forms are shown 
which are intended for feed pumps. 

a. b. 



At a is shown a pair of conical valves. The upper valve and 
seat are made of bronze to avoid rust. The lower one, which is 
the suction valve, has an iron seat. If it is desired to provide a 
bronze seat for both valves they may both be made the same 
size and bevel. The width of bearing, s, may be made as in 
formula (401). If the horizontal projection of the seat is made 



■ o.i6" = \//J 



(402) 



the smaller valve will have a sharper bevel than the larger one. 
In designing the valve chamber, it is important to proportion 
the space over the valves so that the return flow of water shall 
be high enough over the valves to insure their closing, as it is 
possible for the return flow to get under the valves and hold 
them up from closing.f The valves here shown are made with- 
out any packing material. 

At Fig. 11315 is shown a ball valve. In this the width j of 
the seat, and also its projection .j, are the same as in the pre- 
ceding. The diameter of the ball is found by drawing lines at 
right angles to the bevel of the seat from the middle of its 
width, the intersection of the lines giving the centre of the 
ball. X The high position of the outlet opening is necessary in 
order to maintain a proper lift to the valve and keep the seat 
in good condition. 

In order that the opening through the valve shall be equal to 
that of the pipe the lift, /;, of the valve must equal % D. (See 
I 369)- 





Fig. 1 132. 

Disk valves are often made with soft packing upon the seat, 
two examples being given in Fig. 1132. That shovtm at a is a 
valve for a mine pump, packed with leather. The ribs are 
shaped so as to form a cylindrical guide for the valve, this con- 
struction being also frequently adopted for conical valves. At 
5 is a disk valve with rubber packing, similar valves being used 
on many of the Gaskill pumping engines ; all the metallic parts 
are made of bronze.? In many instances disk valves are made 
in the form of a ring, the seat being in two positions, the bear- 
ing being on both the inner and the outer edge of the ring. 





Fig. H31. 



1133. 



Fig- 1133''^ shows the valve for the air pump of a Corliss 
engine at Creuzot. In this case the valve is made of a hard 
material instead of a soft one. The seat is made as usual, and 
the valve is a ring of phosphor bronze, held down to the seat 
by a strong flat helical spring. The form shown at b is another 
style of ring valve much used in the air pumps of English 
marine engines. 



•Chief Engineer of '^ Fonts et Chaussees," of France. The subject of 
weirs and movable dams has been very skillfully worked out by French 
engineers. 



tSee Zeitschr. Deutscher Ingenieure, 1S86, p. 97. 

i See Uhland, Prakt. Maschinen Konstrukteur, 1870, p, 83. 

g See Engineering and Mining Journal, April, 18S6, p. 385. 



Plate 14. 



2/6 



THE CONSTRUCTOR. 




Fig. 1 134 is a so-called 
"bell" valve, used in 
mine pumps. Here the 
two seats for the ring of 
the valve are in different 
planes. The seats are 
packed with oak with 
the end grain up. The 
outlet in this form is 
around both the inner 
and outer bearings, in 
which respect it differs 
from Fig. 11336. The 
lift p, which is required 

7r 
to give an area of — Z?^ 
4 

is somewhat less than 
before, being equal to 

I EP 



Fig. 1 134. 



4 Z?i + A 

The necessity for lim- 
iting the lift of valves in 
pumping machinery has 
led to the use of a large 



number of small valves in the same valve chamber in order to 
obtain the required area with small lift. 

A distinction may be made between two methods of arrang- 
ing such valves. The first method consists in arranging a 
number of similar round disk valves each over its own opening 
in a plate. An example of this is seen in Fig. 1016, in which 
rubber valves similar to Fig. 11326 are arranged in rows. The 
phosphor bronze valve. Fig. ii33«, is also used in this manner, 
38 being placed on the suction side, and 27 on the discharge 
side of the air pump. 

In a round valve chamber the arrangement of the valves is 
more difficult, both as to the placing of the valves and to pro- 
vide guides to control their lifting and seating. 





Fig. 1135. 

Fig. ii35rt .shows a set of 19 valves as used in the Heidt shaft 
at Hermsdorf, and Fig. 11356 a set of 21 ball valves in the 
Joseph's shaft at Frohnsdorf* The,se are both shown inde- 
pendently of the casing. This system has shown itself so 
advantageous that it has been extended until sets of several 
hundreds of ball valves, acting as a single valve, have been put 
into use. Fig. 11356 shows one feature which must always be 
taken into account, namely, the relation which the size of the 
valves and valve casing bear to the water pipe. In this instance 
the diameters of the casing and pipe are igj-^ in. and -/% in., 
and the areas as 7.4 to I. 

The second method of arranging a number of valves is sug- 
gested by the bell shaped valve of Fig. 11 34. In this case the 
stream which flows toward the centre is above the one which 
flows outward, thus providing sufficient room for the flow of 
the upper stream. This idea is also used in the arrangement 



shown in Fig. 1 1356, the inner circle of balls being placed 
higher than the outer circle. By extending this idea of super- 
posing the discharge openings of a number of valves we obtain 
a construction consisting of a number of ring valves, forming 
what may be called a set or cone of valves,t of which three dif- 
ferent forms are shown in Fig. 1 1 ^6. The form shown at a is 
used in the large pumping engine of the Scharley-Tiefban 
mine, t the pumps being i metre diameter (39.37 in.). This 
consists of a number of ring shaped valves of constantly dimin- 
ishing diameter, constructed on the bell principle, the seat of 
each valve being on the one next below. 




Fig. 1 1 36. 

The form at b is the design of Thometzek,§ and is very prac- 
tical. The ring valves are all alike in size and form, each hav- 
ing its own seat, these being built up as high as may be required 
and held in place bj- a screw bolt through the lid of the valve 
casing. 

The design c is that of the Humboldt Machine Work at 
Kalk. II The ring shaped valves of bronze are slipped over the 
succession of seats which form a cone of stepped shape, also of 
brouze. These seats, as in the system of Thometzek, are sep- 
arate, and are held together by a screw bolt on top, with the 
difference, however, that each valve in lifting strikes against 
the next, the amount of lift increasing in an arithmetical ratio 
from above downward, the uppermost valve being held down 
by a spring. In this last construction the ratio to D is some- 
what smaller than in form b. All of these designs are intended 



* See Riedler, ludikator versuche, etc., p. 27, and plate 11. 



t German " .S/w/^rwr/fM/z'/,'' French ^^ Etagenventile.'^ 

T See Riedler, Indikator versuche, p. 21. 

\ To the best of the author's knowledgfe Director Thometzek, of Bonn, 
was the first to use ring valves arranged in steps (1S75), and his designs 
have been widely and snccessfuUj- used in practice. 

\ A very good summary of such valves is found in an article by Engineer 
Waldastel, entitled, " Ueber Ringventile fiir Pumpen uud Geblase," in Z D. 
Ingenieure, 1SS6, p. 935. 



THE CONSTRUCTOR. 



277 



for water pumps, br: an excellent form is designed by the 
Humboldt Machine Works for blowing engines also, the suc- 
tion and discharge valves being concentrically arranged.* 

I 368. 

UnbaIvAnced Pressure on Lift Vai^ves. 

If we assume the joint of contact of a lift valve to be entirely 
tight and represent the projected area subjected to the pressure of 
the discharge column by F^, the area exposed on the under- 
side being called F, we have at the instant of equilibrium of 
the two columns as the valve is about to lift, p F ^=p^ F^, in 
which p and />, are the pressures per unit of area on each side, 
and the weight of the valve is neglected or counterbalanced. 
From this we have 






F ' 



or of the ratio -~ is put : 



A 



(403) 



The pressure/ — p^ is the unbalanced pressure on the valve, 



and the ratio -^ 



P-P, 



is the ratio of unbalanced pressure. 



Upon this question of unbalanced pressure much depends, 
and many calculations have been made for various sorts of 
valves, the pressure tending to close the valve being much 
reduced in bell shaped valves, such as shown in Fig. 1134. 
Experimental researches, made upon pumps of various sizes, 
however, have shown that only a small excess of pressure is 
actually required, t At the same time the preceding formula 
shows that the question of the unbalanced pressure is by no 
means a subject to be neglected.! 

As an instance of the effect of unbalanced pressure may be 
cited a bell shaped valve, i metre clear opening, in the shaft of 
the Bleyberg mine, of which the seats could not be kept down 
by their own weight, but would adhere to the valve, rising and 
falling with it until secured by some other means. 

Riedler has observed the fact that in arranging valves in a 
series in a cone as iu Fig. ii36(Z, the uppermost valve which is 
subjected to the greatest excess of pressure according to (403), 
lifts first, and is followed by the others, the lowest rising last. 

It appears that a thin film of water is retained between the 
bearing faces of valve and seat, which responds rapidly to the 
pressure of the lower column p^ ,and thus tends to reduce the 
value given by the above equation. If we first make the 
assumption that such a film exists and acts in the manner indi- 
cated, we have for two successive ring valves, arranged for 
example as iu Fig. 1136a, the following stresses in the liquid. 
The weight of the valves, beginning from the top, is indicated 
by G-^ and G.^, and their projected areas by Fy and F.,. 



P'=p,-\- ?i and /" = /. + -^;^-% 



(404) 



Now it appears by examination of the weights and areas that 

G G 

txnder the circumstances ^^ is greater than — ', which is then 
F„ F^ 

also true for the entire second member of the value of /" |, so 
thatj*' is the resistance which is overcome first. In the case of 
the Bleyberg mine F^ is very much greater than F^, and p'^ 
becomes less than p' which explains tne action of the yalve 
seat. 

The actual behavior of the film of liquid between the surfaces 
of contact may not be so definite as indicated above, but it ap- 
proaches to it as an approximation. This is shown by the 
very valuable researches made by Prof. Robinson upon a valve 
acting under steam pressure. || In two extensive series of ex- 
periments he investigated the actual weight required to lift a 
valve under pressure. The results showed that the unbalanced 
pressure was much less than py — p. 



♦German Patent, No. 33,103, 

t Reference is especia'iy made to the numerous and valuable investiga- 
tions of Prof. Ried'er. 

X See the comprehensive papers of Prof. C. Bach, in Zeitschr. D. Ing. for 
i386. " Versuche zur Klarstellung der Bewegung Selbstthatiger Pumpen- 
ventile.'' 

§ In the case of the arrangement shown in Fig 1136a, the ratio of weight 
and area for the three valves, proceeding from above downwards, is 50 : 76 ; 
SS- 

II See Trans. Am. Soc. M. E-, Vol. IV, 1832-1883, p. 350. 



r-a* 



The experimental valve, shown in Fig. 1137, had an annular 
seat of 6 in. outside and 2^^ inside diameter, and was subjected 
to a steam pressure />, above, 
and to the atmospheric pres- 
sure/ below. In the follow- 
ing table p' indicates the 
pressure per square inch 
which would give the equiv- 
alent of the actual pressure _ 
F required to lift the valve, V- 
while a is the area and d the 
diameter of a circle for which — 
^ iPi — P) = ^- This circle 
Robinson calls the circle of 
equilibrium, and it is always 
smaller than the upper pro- 
jection of the valve. 

The valves under a and d 
are taken approximately at 
the nearest values. The un- 
balanced pressure can readily 
be determined from the table. Fig. 1137. 



P- 



i'x-p 


/' 


a 


d 


d' 


Pounds per 


Pounds per 


Square 






Square Inch. 


Square Inch. 


Inches. 


Inches. 


Inches. 


5 


8 


5-^ 


26 


2.53 


10 


17 


5-8 


2-7 


2.85 


15 


26 


6.0 


2.8 


2.92 


20 


36 


6.2 


2.8 


3.02 


25 


46 


6.4 


2.9 


3-09 


3° 


57 


6.6 


2.9 


3-14 


35 


69 


6.8 


2.9 


319 


40 


Si 


7.0 


30 


3.22 


45 


95 


7-3 


30 


3-25 


5° 


112 


7-8 


3-1 


3-27 


55 


129 


8.2 


3-2 


3-29 


60 


150 


8.7 


3.3 


3-31 


65 


172 


5-; 


34 


3-33 


70 


1 98 


9.8 


3-5 


3-34 


75 


230 


IO-5 


37 


3-35 



If />! — / = 45 lbs. we have, since aT = 3 in. = )A 6 in. for the 
excess pressure, one-fourth /, — p ; for p^ — / = 75 lbs. it is 
equal to 0.3S (/>, — p). The law of reduction of pressure 
between the surfaces from p^ to p is not simple. The corres- 
ponding curve is convex towards the axis of abscissas, as shown 
in Fig. 1 137. If it is desired to determine the mean pressure 
pm we have from the table for p^ — p =1 ^ the value 



pm '■ 



A— A 

4-43 



forpi — /■ = 75 it is/„ : 



A-/_ 

2.36 



For a rough 



approximation we may put/>„ = }{ (p^ — /). Prof. Robinson 
has deduced a theory from these experiments. He assumes 
that between the surfaces there exists between the pressure A 
at the outer circumference to the pressure p, at the inner cir- 
cumference, a gradual increase of pressure from/ to/,. Under 
the assumption that the fluid under consideration is incom- 
pressible he obtained by pure analysis the following equation 
for the value of d : 



0" = 2 r 



./ 



R 



(^-0 



(405) 



in which I? and r are the inner and outer radii of the ring of 
the seat. The values of d' as obtained from this equation are 
given in the fifth column of the table. They increase nearly 
as the experimental determinations of d, but with Robiuson's 
assumption of an entirely elastic fluid they are 10 to 15 per 
cent, too great. Probabl}' steam should be considered as mid- 
way between an elastic and a non-elastic fluid. 

The deductions from Robiuson's experiments are hardly ap- 
plicable to pump valves because the lifting of the valve by the 
action of the lower column is effected by a varying pressure, 
while in the experiments / was uniform. If we accept Robin- 
son's theory we arrive in fact to what has been already stated, 
namely, that when the value of/ increases between the surfaces 
until it reaches /,, the pressure p., will be balanced, since in 
equation (405) for / — /, the value of a" r= 2 r, that is, the 
unbalanced pressure becomes zero. This also agrees with 
Riedler's indicator tests, since experiments with the indicator 
failed to show appreciable unbalanced pressure. 



278 



THE CONSTRUCTOR. 



These experiments appear to indicate that practically the 
unbalanced pressure cannot be great, and in most cases for self- 
acting valves it may be neglected. Prof Robinson's experi- 
ments and theory may serve to determine with considerable 
accuracy the pressures at which a safety valve begins to lift. 

I 369- • 
C1.0SING Pressure of Self-acting Valves. 

As already shown, a self-acting valve opens whenever the 
pressure in the under column exceeds that above the valve. As 
soon as the direction of pressure is reversed the valve should 
close quickly. This is especially important, as Riedler has 
shown in the case of suction valves, since when the closing is 
delayed appreciably after the reversal of the pump piston, the 
moving column of water is checked with a sudden shock. For 
this reason the suction valves are given especial attention, as 
shown in the example already cited from Creuzot, in which 
there are 38 suction valves and only 27 discharge valves. 

In order that the lift 
shall not be too great and 
to insure prompt closing, 
the valve may be loaded 
with a definite pressure, 
K, obtained either from 
the weight of the valve, 
or by means of a spring, 
or by both. This ques- 
tion will here be exam- 
ined. Referring to Fig. 
113S, we have for the 
lifting pressure due to 
the under column : 




P- 



= A + 7?^ = A + ? 



(406) 



in which /> — P\^= q the closing pressure per unit of area. For 
a height //, and putting 11 = the circumference of the cylindri- 
cal space inclosing the valve, we have : 

<£', /: :'. = Fv 

■Wi being the velocity of flow at the outer edge of the valve, 
and t' the velocitj' of flow in the under column, h being in feet. 
Now if 2V is the velocity at the inner edge of the valve we have 



that is : 



But we also have 









W= ^ 2 gh' = ^2^X2-3? 

(since the pressure per square inch is equal to ^ — 1 and hence : 



Substituting, we get : 



2 ^ X 2.3 ^ 



Fv 



Now it is desirable that -v and zc, should not be too great ; 
that is, the ratio of h u to i^ should be equal to, or less than, 
unity. If we put h u — j3 F, we have : 



/5 



4- 



2.5" X 2.39 



and, putting for g its value ^ 32.2, we get : 

a iP' a v' 



or say = 



* 148.12 ^2 

from this formula we get for : 

424 
g = .006667 a v^ .01185 a z^ .02666 a v' .1066 a v' 



(407) 



/3= I 



in which v is at its maximum value when it equals the velocity 
of the pump piston. For purposes of numerical calculation we 
still require the value of n. Taking the width of bearing s, and 
projection in the case of conical valves s^ from (401) and (402) 
we have : 



Dia. D = 


2 in. 


4 in. 


6iii. 


8 in. 


10 in. 


12 in. 


16 in. 


Width of seat s = 
Projection .ri = . . 
Cone valve a = . 
Flat Valve a = . . 


0.44 
0.28 
■■65 
2.17 


0.56 
0.40 
1.44 
1.64 


0.64 
0.48 
1.36 
1.44 


0.72 
0.56 
I 31 
1-39 


0.80 
0.64 
1.27 
'■35 


0.84 
0.68 
1.24 
1.30 


0.96 
0.80 
1.21 
1.25 



An example will show how the pressure of closing can be 
calculated : 

Example I. — For a conical valve -whose smallest diameter Z* = 4 inches, 
and the greatest velocity v of the lower column is 6^^ feet per second the 
area of inlet of valve hu = F^ and /3 = 1, we have a pressure of ^ = .006667 X 
1.44 X (6.5)" — 0-4 lbs. per sq. in. For the total pressure we have 

K— (4 + 2 X o-4°)- X 0.4 = 7.24 lbs. 

4 
Example 2. — For a fiat valve of the same dimensions we have a = 1.64 — 

whence K = — - — 7.24 = 8.24 lbs. 
1.44 

The method of calculation is similar for ring shaped valves 
and can readily be applied. The formula (407) can only be 
considered as an approximation as the variations in the jet of 
water affect the pressure. It is evident, however, that K is 
often quite large. 

In the preceding calculation the momentum of the water 
column has not been taken into account. In some cases this is 
sufficient to hold the valve open until the piston has made a 
great portion of its return stroke. This is well shown in the 
case of the pump at the Bleyberg mine (? 319, note) which ap- 
parently showed a discharge of 104 per cent. If this action can 
be made to exist during the entire stroke by giving the water 
a sufficient velocity by contracting the tube that the discharge 
valve does not close at all, this valve may be entirely omitted. 
This is the case with the single valved pump of Edmond 
Henry,* which has only a suction valve and no discharge 
valve. An analogy to this form of fluid ratchet is found in 
Langen's fly wheel ratchet train. Fig. 730 and 731. In this case 
the momentum of the fly wheel is sufficiently great for it to 
suffer no perceptible loss of velocity during the return stroke 
of the pawl. 



Mechanically Actuated Pump Valves. 

The numerous investigations of recent years have show. ' 
that by proper loading of the valves, combined with a reduc- 
tion of lift, the shock of the water in a pump can be verj' ma- 
terially reduced and kept within practical limits, even for high 
piston speeds. The reduction of lift involves a great multipli- 
cation in the number of the valves and a great increase in 
dimensions. For this reason another solution of the problem 
has been attempted, namely, that of abandoning the self-acting 
feature, and actuating the valves by mechanical means. The 
best arrangement seems to be that in which the valves are 
opened by the action of the water, but closed by a positive 
gear in advance of the shock. The application of this method 
enables the size of the valves to be reduced, and as it is princi- 
pally used for large pumping engines the valves can be oper- 
ated by connection to the flj' wheel shaft. Professor Riedler 
has recently made very valuable investigations upon this 
system, t 




Fig. 1 139. 



Fig. 1 139 shows the valve gear for the Riedler pumping 
engine at the Wartinberg mine. The revolving cam d, closes 



* See Revue Industrielle, p. 342, September, 1SS8, where the complete 
theory of this form of pump is given. 

fSee Riedler, Mine Pumps with Positive Valve Gear, Zeitschr. D. Ingen- 
ieure. 1S88 p. 481. 



THE CONSTRUCTOR. 



279 



the valve b, just as the plunger is at the end of the stroke, and 
permits it to open by the action of the water. The valve is 

held to its seat by a 
a spiral spring. Pumps 

of this construction 
operate very smooth- 
ly. Further details 
of this construction 
are given in the arti- 
cles already cited. For 
blowing engines, and 
especially for air com- 
pressors, positive ly 
actuated 'valve gears 
are much used. A 
very simple action for 
the inlet valves is 
shown in Fig. 1140. 
The piston rod c 
moves the valve b, 
by means of the fric- 
tion of the rod in the 
stuffing box, the ac- 
tion taking place just at the reversal of the stroke. Examples 
of this construction are to be found in the air pumps for use in 
physical laboratories. 

? 371- 

Valves with Spir.^i, Movement. 

It is not so convenient to construct a valve so that its motion 
shall be both rotary and rectilinear axially, and this construction 
is mainly limited to valves which are operated by hand. 




Fig. 1140. 



This is the counterpart of the throttle ratchet shown in \ 250, 
and valves of this sort have been much used with throttling" 
governors for steam engines. The closing of such valves is im- 
perfect, as the edge must be rounded near the hub of the valve, 
thus giving only a line of contact* 





Fig. 1142. 

If it is desired to use throttle valves for regulation of water 
pressure, as the case of turbines, etc., it must not be forgotten 
that the resistance of the valve will materially affect the effi- 
ciency. 

For self-acting valves a variety of throttle valve may be used, 
in which the area of one wing is only about )-l to Ys, that of the 
other wing, thus partially balancing the valve. This form, 
which is old, appears to be again coming into use. t 

Lift valves which are situated in vessels which are not closed 
at the top may be balanced in a simple manner by making the 
valve with a tubular continuation which extends above the sur- 





FiG. 1141. 

Fig. 1141a shows a conical valve with spiral motion, as used 
on the Giffard injector. This arrangement enables a very fine 
adjustment of the opening to be obtained ; a similar form is also 
used in the so-called " cataract " for steam engines. The sharp 
point of the cone has caused valves of this sort to be called 
"needle" valves, and similar forms, without the spiral action, 
are found in gas regulators. Stop valves for steam and for 
water are frequently made with spiral motion. An example is 
shown in Fig. 1141^. When the valve is not in contact with its 
seat it has both a vertical and a rotary motion. In the parti- 
cular form shown the valve has a disk of asbestos which forms 
the surface of contact with the seat. This general form is 
known as a "globe " valve on account of the form of the body, 
and such valves are very extensively used for steam and water. 




? 372- 
Bal.anced V.ai,ves. 

Valves which are to be operated by other means than by the 
action of the fluid, are advantageouslj- made so as to be relieved 
from fluid pressure, and thus offer less resistance to operation. 

Valves of the wing or flap construction are conveniently bal- 
anced by combining two valves moving in opposite directions 
into one valve of the form commonly called "throttle" valve. 
Fig. 1 142. 



Fig. 1 143. 

face of the water. A balanced valve upon this principle, as 
used for an outlet valve in a canal lock, as at b-^ and b./ , 



*The form shown at 5 is recommended in Revue Industrielle, p. 205, May 
26, 18SS, as insuring a better balance, but from Robinson's experiments, 
already cited, this form would offer too much resistance to opening. 

tSee Belidor, Architecture Hydraulique, Paris, 1739, Vol. II. These 
valves were of brass with metallic packing. 



iz8o 



THE CONSTRUCTOR. 



"Pig- 993> IS shown in Fig. 1143. This valve, designed by 
Constructor Cramer, is made with a cylindrical shell of sheet 
iron extendmg to the surface of the water. The diameter of 
this shell is the same as that of the valve, and the weight of 
the valve, which is by no means small, is partially counter- 
balanced, leaving only sufficient to insure proper closing and 
seating. * _ If it is desired to apply Cramer's construction to 
valves which are subjected to high pressure, this may be done 
by using two stuffing boxes, one external and one internal, 




Fig. 1 144. 

as shown in Fig. 1144, which, however, adds to the complica- 
tion. For lift valves which are to act under high pressure a 
better construction is the so-called "double-beat" valve, 
which, like the throttle valve, consists of two similar valves in 
which the pressures oppose and neutralize each other. Three 
forms are shown in the accompanying illustrations. Fig. 1 145a 




Fig. 



1145a. 



being a double disk valve, and Fig. 1145,5 a tubular valve. Both 
of these were invented by Horublower in the latter part of the 
last century. Fig. ii45(; is a bell or Cornish valve. These 




Fig. 1 1454. 

valves each consist of a pair of conical lift valves, the varia- 
tions appearing in the details of the connections and passages. 

•See Annales des Fonts et Chaussees, 6me serie, Vol. XII, 1886, 11 Semes- 
tre, p. 248, also Zcitschrift fur Bauwesn, 1880, p. 155. 



When the projection of one seat falls within that of the other, 
as in forms b and r, the unbalanced pressure is that due to the 
projections of both seats. If so desired, however, these may be 




Fig. 1145c. 

made as Fig. 1145(7, with one seat directly over the other, in 
which case the pressure/; — p need only be calculated for one 
seat. For the preceding double seated valves we may make : 



for the width of seat i.= '/,' (0.2 »Ad -\- 0.137 I 



and for the projection j-j = yi 



0.2 ^D''^ 



(40S) 



In form a the mean diameter D' of the valve is = o.S times 
the diameter D of the pipe, while in forms b and c the diame- 
ters of valve and pipe are the same. For the force required to 
lift the valve, taking the projection jj into account and assum- 
ing the pressure between the surfaces to be as in \ 36S, equal to 
Vi {Pi P), we have, neglecting the weight of the valve : 



P' 



IT D' s,' 



VyiiPi-p) ■- (409) 

while for a single conical valve of the same diameter D it would 
be : 



P = 



G 



D'- + =.3' s,^{D + s\) 



](A-; 



• (410) 



P is proportionally very great, while P' is not always unim- 
portant. 

Example.— For D' — 12, we have for form a, Si' = yi, { 0.2 ^/Ta ) =0.346". 
If uow />! — p = 60 pounds per square inch we have ; 

-f = IT X 12 X 0.346 X 7^ X 60 = 521 pounds. 

For a single valve the diameter would be Z* =: — -- = 15 inches, and from 

'402) Si = 0.2 \/ 15 = 0,77, whence 

P= -— 15- + -A X 0.77 IT ( 15 -(- 0.77 j 60 = 12,126 lbs. 

so that F is nearly 24 times P'. 

It is verj' desirable for double seated valves which are to be 
used for steam, that both valve and seat be made of the same 
material, in order to avoid unequal expansion. 




Fig. 1 146. 

Double seated valves are also used for water, 
shows such a valve arranged for a sluice. 



Fig. I 146 



THE CONSTRUCTOR. 



281 



This valve is made with flat seats, the lower seat being faced 
with rubber, and the upper one packed with leather secured to 
the housing whicli is shown over the valve. The valve red runs 
through this housing and through a tube above the surface of 
the water. The diameter D is 1400 mm. = 4 ft. 7 in. This is 
practically a tubular valve, similar to Fig. 11456, except that 
the direction of flow is reversed ; this arrangement has also 
been used by Hornblower. The leather packing at 2" is made 
flexible, since the projections of the valve seats lie one within 
the other so as to make a slight tendency for the valve to lift, 
without entirely overcoming the weight of the valve. Balanced 
valves of the kind described above are also adapted to large 
steam engines. In some instances a small balanced valve is 
arranged so that it is lifted first and admits steam under the 
main valve before the latter is lifted. 

Another device is that shown in Fig. 1147, known as Ait- 
ken's automatic steam stop. The main valve b, is closed by 
being screwed up against its seat by the spindle and hand 
wheel. Before opening, it is balanced by admitting steam 
through the by-pass vah-e b' . The valve itself is loose on the 
spindle, and if through any breakage in the pipe beyond the 
valve a sudden or rapid flow of steam should take place, it will 
be automatically closed by the force of the current. 



plug and one for the spindle. The management of screw cap 
and jam nut enables a fine adjustment to be obtained.* 




Fig. 1 147. 

Lift valves may also be balanced by making a balance piston 
connected with the valve, the pressure of the steam acting upon 
the piston in the opposite direction to the action on the valve. 
This construction has also been applied to reducing valves in 
the place of weighted levers or springs in various ways, but 
space cannot here be given to the subject. 

B.— SLIDING VALVES. 

I 373- 
Rotary Valves and Cocks. 

For rotary valves the bearing surfaces are conveniently made 
conical, so that a simple endlong pressure on the valve will 
hold it firmly to its seat. Valves of this construction are 
known as cocks. 

Fig. 1 148 shows two forms of such cocks which are in general 
use. The opening through the plug of the cock increased in 
height in order to obtain a full area without requiring the 
diameter of the plug to be too great ; the area of the opening 
through the plug being made equal to the area of the pipe, i. e., 

= -D\ 
4 
According to the experiments of Edwards, a good taper for' 
the plug is \ on each side. For the thickness i of the metal in 
the body of the cock formula (319) may be used when the 

D 

material is of cast iron, which gives & ^ 0.472'' -| ; for bronze 

50 

the thickness ma}' be made one-half to two-thirds this value. 
The design shown in Fig. 114814 has the plug entirely inclosed 
in the body, and is made with two stufiing boxes, one for the 




Fig. 1 148. 

Fig. 1 149 shows two forms with hollow plugs, these being 
much, used for injection cocks for jet steam condensers. 





Fig. 1 149. 

When the angle of the apex of the cone becomes iSo° the 
plug becomes a flat disk, and this form is often found in the 
throttle valves of locomotives, and less frequently in the valve 
gear of engines. True cylindrical plugs, i. e. , those in which 
the angle of taper is equal to zero, are rarely used, although 
recommended by some. This form is better made in a portion of 
a c^'linder, and operated by an oscillating motion, as in the 
Corliss and similar valves. A starting valve of this type, used 
as the steam admission valve for a triple expansion engine is 
shown in Fig. 1150. 




Fig. 1 1 50. 

At a is a longitudinal section, b a cross section, and at c is 
shown the seat looked at from above. In the one seat three 
passages are controlled at /', I" and /"'. All three are closed 
when the valve is in the position shown at b, but open at the 
same time when the valve is moved to the left. Tlie trapezoidal 
opening in /' admits a small amount of steam to the high pres- 
sure cylinder at the same time that a little live steam is admit- 
ted through /" and /" to the intermediate and low pressure 
cylinders, so that the engine is sure to start. The valve is then 
thrown all the way over, closing /" and /■"' and throwing /' 
wide open.f 



* Mosler's German Patent, No. 33,912. 
t See Zeitschr. D. lugenieure, p. 509, 
Marine Engine. 



Meyer, Triple Fxpansion 



282 



THE CONSTRUCTOR. 



i 37-1- 
Gate Valves for Open akd Ci.osed Conductors. 

A great variety of valves has been devised for open water con- 
ductors in the form of gates bj' which the flow can be regulated. 
Such gates have been preferably made of wood with the excep- 
tion of the operating mechanism. At the present time iron is be- 
ginning also to be used for the gates, and as in the case of other 
branches of work, wood is likely to be less and less used, being 
limited to a few special cases. For very broad streams the con- 
struction of such gates is now sometimes made upon the princi- 
ple of subdivision. In such cases the breadth of the stream is 
subdivided into a number of smaller streams, each with a sep- 
arate gate, thus keeping the gates small enough to be m.ovable 
by hand. 

A weir which is placed in a stream is both in principle and in 
construction a valve. When the water in the stream is low the 
flow is entirely checked ; for the mean flow the stream passes 
through the reduced opening with a velocity due to the reduc- 
tion in section, while for high water the entire width of the 
dam is overflowed. Movable weirs are plainly examples of 
regulating valves. French engineers have given much atten- 
tion to moveable dams with excellent results. A new design 
for a moveable dam by Schmick is shown in Fig. 1151.*. This 
dam consists of a number of pontoons, each three of which are 




Fig. 1151. 

secured together by a yoke and anchored by a chain to a point 
up the stream. All three pontoons of each set are arranged 
with variable water ballast in two or more compartments, a/ 
and a./. An adjustable valve 3, enables comnranicatiou to be 
made with the upper water level, and the compartment a/, and 
a similar valve b., connecting the compartment a./ with the 
lower level, while a third valve b., enables communication to be 
made between the two compartments. By varying the open- 




FiG 1152a. 



ings of the valves the pontoons can be caused to regulate the 
difference of water level above and below the pontoons, while 
if all three valves are closed the pontoons will rise and fall with 
the variations in the level of the stream. 

Gate valves are much used for water mains, and an example 
of the many varieties used for the purpose is shown in Fig. 
1152a. The gate or disk of the valve is made of bronze, and is 
wedge shaped, in order that it may be firmly pressed against its 
seat "when the screw is tightened (this forms a pressure of the 
second order) while the pressure is immediately relieved at the 
commencement of opening. The screw is in this case made of 
' ' sterro-metal ' ' to avoid rusting. 




Gate valves are also used for gas mains, and a valve for this 
service is shown in Fig. 'i\^2b: In this instance the valve is 
operated by means of a rack and pinion. The motion is made 
in the horizontal direction so that the valve will remain in any 
position, the only resistance being that of friction. 

? 375- 

Sl,IDE Vai<ves. 

Slide valves are mainly used for the purpose of effecting the 
distribution of steam in steam engines. This is such an im- 
portant subject that al) the forms in general use will here be 
noticed. 




Fig. 1 153. 

I. Plain D valve, Fig. 1 153. This is the most important 
form of all. The action of this valve has already been discussed 
in I 328, and hence the dimensions will only be considered 
here. The width a of the steam ports is kept as small as is 
practicable, w^iile the length at right angles to the plane of the 
drawing is made quite large. When a is given, the dimensions 
to be determined are the outside and inside lap e and i, the 
bridges b, the width of face bo beyond the ports, the width ao of 
the exhaust port IV, the travel r, the length of the valve /, and 
of the valve seat h. The laps e and i, and under some circum- 
stances two valves <?., and e^ for e are determined according to the 
method given in Figs. 1024 and 1025. In the same manner also 
is found the greatest distance.?. Fig. 1153d, in which the edge 
of the valves passes the edge of the port. This gives the width 
of bearing t of the valve upon the bridge, since b = s + t. The 



value of t varies greatly, the least permissible value is t 

and it is more frequently made y%" to %". Approximately, for 
we have, after assuming t as just given, ao -\- t — [e -\- a -\- i) 
=- a, in which c is taken as a mean between e^ and e^. We then 
have : 

ao = 2 a + / -\- i — t -j 

whence r = (Z -f £■+ -s f 14") 

and ;r=4fl + 3/+z'+2.j+/J 

The valve face must have an inner width of bearing to Fig. b 
at least equal to t, whence for the total width of the valve face 
we have the value 

ao -\- 2 b -]- 2 a + 2 bo,or: . \ 

lo = ^a-\-2,e — i + A^ + t+2to!i 
The thickness of metal in the valve itself, when made of cast 



16' 



(412) 



*S. Schmick, Prahmwehr (Pontoon Dams), Zeitschrift fiir Bankunde, 
Munich, 1S84, p. 502. 



3n should be about = 



r + o-V 

200 



vhich is about half the 



THE CONSTRUCTOR. 



283 



thickness of the metal, of the steam cj'linder as given by formula 
(3JO). If the valve is faced with white metal the body of the 
valve should be of bronze, the white metal itself not being strong 
enough. 

,.e«-a^i---s+t+atr-*tH 2. Double D valve, Fig. 

1 154. In this form the four 
valves which in the plain 
D valve are united in one 
piece, are separated into 
two portions, connected 
by a rod. This construc- 
tion is adopted to shorten 
the steam passages / and 
///, the ^width of each 
valve is = 3 (2 -I- ie -\- ^s 
■\- t -\- to ox oi both to- 
gether ^l^=i,a-\-i,c-\-^s\-2t-\-2to. 

3. Pipe Valve, Fig. 1155. This form is also intended to re- 
duce the length of the ports // and ///, which is often an im- 

Hi-su + a+r^-aji^ 



Example 2. — For 




FiG, 1 154. 





Fig. 



155- 



• b, ao b "a to; 

stroke. The total 
+ 5« + 3-?+'' + 



portant consideration in engines of lon^ 
length of valve bearing surface is / = 6 ( 
2 to. 

Example I.— If a = y^" , e = 14, i = i|", i = ^", i = i^ = ^we have for a 
plain Z) valve the width / = 4 X 0.75 -|- 3 X o.75 -t- 0.6875 -)- 2 X 0.375 -|- 0.1875 
^ 6.875". 

For the double D valve we have / = 6 < 0.75 -t- 4 X 0.75 + 4 X 0.375 + 4 X 
0.1875 = 8.75" and for a pipe slide valve as Fig. 1155, ^= 6 X 0.75 + 5 X 
0.75 4- 3 X 0.375 + 0.6875 + 2 X 0.1S75 = 10.4375". The work of friction in 
moving the valves is directly in proportion to the above widths, since the 
travel is the same in all three cases, being : 2r=2e + 2rt-|-2^ = 2X 0.75 
-I- 2 X 0.75 + 2 X 0.375 = 3-75". 

In order to reduce the work of friction in .slide valves the 
multiplication of valves has been resorted to, much as has 
already been shown in the case of lift valves. A division of the 
valve system into two parts has also been made for marine 
engines with oscillating cylinders, the object being to place one 
portion on each side of the cylinder and thus keep the entire 
mass symmetrical with regard to the axis of oscillation. In 
this arrangement the two slide valves correspond to eight sepa- 
rate valves. In these as also in engines, with stationary cylin- 
ders, the valves may be combined into one. This may be ac- 
complished by using two or more sets of steam passages which 
unite at one point and by making corresponding divisions in 
valve and valve seat. The combination of several valves so as 
to act as one is not limited to lift valves, as manj' useful forms 
of slide valves are made on this principle, some of the best 
forms being here shown. 

4. Penn's Gridiron Valve, Fig. 1156. In this the steam port 



a is divided into two ports, each having a width ^ 



To de- 




termine the total width of valve as in the previous cases, we 
have : t = 5-5 a -{- ^.5 e + i s + t + 2 to + % i, and for the 
travel : 2 >'= a -]- e -\- s, that is half as much as before. It is 

evident that the laps — and — must bear the same relation to 



— as the diagram gives for a : e: i, in the preceding forms. 

2 



5 
16 ' 



^a 



= ^5- we have / = s-S X 0.75 + 3.5 X 0.75 + 3 x 0.375 + 0.1875 + 2 X 0.1875 
+ 0.3125 = 8 75" 

2 )- = 0.75 + 0.75 -I- 0.375 = 1.875. 

This gives for the work of friction of such a valve as compared with an 
equivalent plain slide valve : 

6.S75 X 375 ^ 
8.75 X 1.875 ' '' 
which is an important gain. 

5. Borsig's Gridiron Valve, Fig. 1157. This is the same in 
principle as the preceding, and differs only in construction, the 

.-.L-l- 




Fig. 1 157. 

exhaust passages being carried on each side of the valve instead 
of above, as in Penn's construction. 

6. Hick's Double 
Valve, Fig. 115S. This ^—: -— fa- 

is intended for use with L'3^^_LJi^iLi 

compound engines 
with parallel cylinders 
(Hornblower and 
Woolf), the ports //' 
and ///' are for the 
high pressure cylinder, ^g 
and //" and ///■" for 
the low pressure cvlin- 
der. The width 7 of 
the valve is : 
^=5 a + sai+ 6e + 
4 s + e^ + t^ X to. piQ I , .3 

Usually ffj is made 
equal to <i, which reduces the value of / somewhat. 

7. Allan's Double Valve, Fig. 1159, is a valve for compound 




• boa'li "I'b,- !'^» I bi'ai-b a' bo 



.a+e+i . . a+e+i 



a+e+i , I 




Fig. 1 159. 

engines with tandem cylinders. The value of / is 

/ = TO a + 7 ^ 4- ^1 -f 6 J -f 2 -f ?i -f 3 /<,. 
This construction not only economizes the work required to 
operate the valve, but also gives a very simple arrangement of 
steam passages. 



The £ Valve, 
Fig. 1 160, is used to 
advantage in place of 
the plain Z? valvcwhen 
the use of a valve gear 
actuated directly from 
the pi.ston rod requires 
that the valve shall 
move in the same di- 
rection as the piston. 
(See Fig. ioo5 and 
looS). This valve con- 



a-l-2r ,a-f2t a-1-2r e 




CTIf^Tn^ 



111, 



i-fr 'a" e+r 



Fig. 1 160. 



sists of two L> valves cast together, and the over travel beyond 
the valve seat gives the admission. 

We have as before : r ^= a -\- e -\- s, and 

^ = i-j-r=a + e+^ + s^ 

i = e+7-=a-\-2e+s y (413) 

ao =^ a J 



284 



THE CONSTRUCTOR. 



This gives for the width of the valve .- 
/ = 3a+2/5+2i(, + 2i'or: "I 

which is considerably greater than for au ordinary D slide 
valve. 



(414) 



16 



tffic 



E^iample 3. — If as in Example 1, we make a = £• — 0.75", 

0,375",/ = 0.1S75", we then have / = 13 X o-75 + 0.6875 + 4 X 0.375 + 2 X 
0.1S75 = 12.3125" against 6. S75'' for the plain Z? slide valve. It will be evident 
that the £ valve is only available for small port widths and small laps, as 
will also be seen in Figs. 1006 and looS. The principal vahie of this valve 
lies in the use of the outer edge of the valve seat as the edge of opening, 
which principle also has a valuable application in the following valve. 

9. Trick's Valve, 
Fig. 1 161.* This is a 
double valve and con- 
sists of one D valve 
over another, with a 
steam passage be- 
tween. As before, 
we have r -- a -j- e 
-{- s, and also make 
bo = 2 1? — t, i. c. , the 
inner edge of the 
outer valve when the 
valve is in mid-posi- 
tion, is at a distance -= e from the edge of the valve seat. The 
consequence is that when the valve is moved a distance equal to 
e, say to the right, the passage through the valve opens to ad- 
mit steam at the same instant as does the edge of the valve on 
the left. This gives a steam admission twice as quickly and an 
opening twice as great as would otherwise be the case. The 
following positions from a to /, Fig. 1 162, will show the succes- 
sive actions, the exhaust ports being omitted for simplicity. 




Fig. 1161. 



yvnr--'-'Y--7-~ , - » , „ ,,„,, ■,,.,-,■,.,. .-,,,., ^ , 



M 







a. The admission is just about to take place both from the 
edge of the valve on the left and through the passage in the 
"■'alve. If we apply Zeuner's diagram (compare Fig. 1025) see 




Fig. 1163. 

Fig. 1 163, we must from the point A, which indicates the port 
opening, double the width given by the Zeuner circle until the 



entrance to the passage in the valve is wide open, as at b. By thus 
doubling the opening in the diagram we obtain the curve A i?,. 

b. From this position on, the opening at the left continues to 
grow wider, but that through the valve on the right does not, 
hence on the Zeuner diagram from this point we return to the 
opening which the regular valve circle gives, to which is added 
the constant opening c ^= B B^=^ C,C^ indicated by the curve 
Bi Cy This continues until the inner edge of the opening of 
the valve passage on the left reaches the edge of the bridge as 
at <-. 

c. As the valve continues to move the passage through it is 
gradually closed, but the steam tort is opened to the same 
amount, and hence the actual port opening remains constant. 
This continues until the position d is reached, when the passage 
through the valve is entirely shut off. This is indicated in the 
diagram by the arc C, D, struck from the centre at I. 

(/. The valve continues to move to the right until it is en- 
tirely upon the bridge, the corresponding portion of the dia- 
gram being the arc D E of the valve circle. 

e. The valve from this position moves on the bridge beyond 
the port until it has traveled a distance equal to s, as shown at 
f, during which time the port opening remains constant, as in 
dicated in the diagram by the arc E E' struck from the centre 
I. From this point the same actions take place successively in 
the reversed order. 

It will be seen that Trick's valve gives a much quicker opening 
and also a much longer duration of the full opening than does 
the plain slide valve. It remains to be seen how these features 
can be used to the best advantage. According to Trick's prac- 
tice this is best done by making the value of j negative, and 

also^/. This makes the port opening from Ci to (T/ in the 

diagram constant, as shown in the diagram. 

In order that the apparent contraction of the ports by the 
change in the sign of .s shall not occur, the value of a is made 
greater than would otherwise be the case. Under these condi- 
tions we have for the exhaust port Oo, the equation : 

ao -\~ t — e-^ — a — « = a — .y, 

in which .j is given the magnitude equal to the distance which 
the edge of the valve is moved beyond the edge of the bridges. 
(See Fig. 11 62/). We then have : 

For the exhaust port, «» = 2 j + c, -|- ?' — s — / | 

For the bridge, b ^ e — <'i + -^ — t \ c^jci 

For the passage through valve r = c — / — t\ 1 V4 0; 

For the total valve, l=\a-\-ii,e—i\-\-i — 3^+;" J 

E-vample 4. — Making s -- 

other data the same as the plain slide valve of Example i, and we have : 

a = 0.75 4- 0.1875 = 0.9375"; e = 0.75 ; i ^ 0.6875 fj = 0.078", whence ; 
f— a + e — s = 0.9375 -I- 0.75 — 0.1875 = 1.5" 



2.2655" 
and 



- / = 2 X 0.9375 + 0.07S -t- 0.6875 — 0.1875 — 0.1B75 



- ei + s — t = 0.7; 
c = e — t- 



— 0.078 -f- 0.1875 . 
■ ^1 = 0.4845" 



-o 1875. 



!=4i + ie — e-i+i — 3S-i-i — 3.ys+3 — 0.078 + 0.6875 —0.5625 4-0.1873 
= 6.9835" 
as against — 6.875" for the plain slide valve, which compares very favorably. 




* This valve was invented and made by Trick at Esslingen in 1857, and by 
Allan in England in 1858-1860 ; in the United States it Is correctly known as 
Trick's valve. 



Fig. 1 164 shows the application of the author's valve diagram, 
already shown in Fig. 1024. The action of the inner portion of 
the valve is the same as with the ordinary slide valve. 

For comparison the following dimensions of an executed 
valve by Trick, are given : 

a = 1.77" (45 mm.), f , = / := 0.07S" (2 mm.), / = 0.216" 
{5.5 mm.). 



THE CONSTRUCTOR. 



2^5 



b = -i." (25.5 mm.), e = 0.846" (21.5 mm.), c = 0.55" 
(14 mm.). 

s = — 0.374 (9.5 mm.), ;- = 2.24" (57 mm.), / = 5.27" 
(134 mm.). 

bo = 1.45'' (37 mm.), lo = 5-83" (148 mm.), i5 = 30°. 

Trick's valve is especially well adapted for use on compound' 
marine engines, and has recently been used in the forms of 
double and gridiron valves, as in Nos. 4 to 7 preceding.* 

l 376. 
Bal-A-Nced Si<ide Valves. 

The resistance to motiou due to the pressure is not so great 
with a slide valve as with a lift valve of the same area, because 
in the former case it is only necessary to overcome the friction 
between the valve and its seat. For large valves, however, it 
becomes so great as to render some method of balancing neces- 
sary. It is desirable that even small slide valves should be 
balanced, as by this means the wear upon valve face and seat 
can be greatly reduced. Balancing is most important for steam 
engine valves, and the following examples belong to this class ; 
for other kinds of service it is unimportant. 

But few researches have been made upon the subject of valve 
friction, but from such as have been made, and for such good 
bearing surfaces as are used, the coefficient of friction may be 
taken at 0.05 to 0.04. 

American engineers, as we have already seen, enter into very 
practical investigations and prosecute them with patience and 
success, and to one of these, Mr. C. M. Giddings, we owe the 
following results.f 

Balanced Valve. — Cylinder ii}^ ' X 10". 



P' = 



i> N 



whence a P' 



. The value 

n 



P' shows the 



Revolutions 
per Minute. 


Pressure 
Pouuds. 


H. P. of 
Eugiue— A''. 


H. P. .V 
for Valve. 


Ratio of 
preceding. 


Ratio 


125 

175 
200 


10 

30 
40 


3 

9 

13-5 




2 per ct. 
r.2 " 
1.4 " 


48 
61 

91 



Unbalanced Valve. — Cylinder 9" x 12". n = 100. 





Ratio 




H. P. by Brake. 


JV' 


Ratio a f 






4-5 


4.5 per cent. 


247 


7.0 


3-5 


245 


8.25 


4.0 


330 


8.9 


6.0 


534 


II. I 


7-3 " 


Sio 



Balanced Valve. — Cylinder <)" x '4 





Ratio 




H. P. by Brake. 


.V 


Ratio a /» 


1 1.4 


1.2 per cent. 


137 


13-5 


II 


149 


14.0 


1.0 " 


140 


15.6 


1.0 " 


■56 



The last column in each of the three tables has been added 
by the author of this work, and is obtained as follows : If N 
and N' are the values in horse power of the engine and of the 
resistance of the valve, v and v' , the corresponding mean 
velocities of piston and valve, and P and F' the force upon 

N' 
each, the experiments give the relation — ^ipor P'v' ^^ Pv. 

Hencejit follows for the force required to move the valve: 

1p P V 



P =■ 



Now for a given engine v' bears a constant re- 



lation to the number of revolutions n, so that we may put 

* See Zeitschr. D. Jng. 1S88, p. 509, Triple Expansion Engine by'G. L. C. 
Meyer, of Hamburg. 

t See Trans. Am. Soc. Mec. Engrs., Vol. VII, p. 631 : C. M. Giddings, Descrip- 
tion of a Valve Dynamometer for measuring the power required to move a 
slide valve at dilTerent speeds and pressures. 




Fig. 1165. 



the 



increase in power required to operate the valve. It is evident 
that /" increases more slowly than the increase in steam pres- 
sure, but the resistance becomes quite great for unbalanced 
valves. The present difficulty lies in the limi'ed number of 
engines upon which experiments have been made. Fig. 1165 
shows the character , 

of diagram made by ' ' 

Gidding's apparatus, 
and it will be seen 
that the greatest re- 
sistance occurs at the 
beginning of the 
stroke, diminishing 
toward the end to 
nearly zero. The 
inequality between 
the resistance of the 
back and forward 

strokes is due to the action of the steam pressure upon 
area of the valve rod. 

In considering the pressure upon unbalanced slide valves ihe 
consideration mentioned already in connection with Robin- 
son's experiments on lift valves is that there exists a counter 
pressure between the valve and seat which overcomes an im- 
portant portion of the pressure on the valve. As a rough ap- 
proximation we may take this pressure between the surfaces as 
■^ (/>! — p). Gidding's experiments show that the coefficient 
of friction is not constant, but diminishes with increased speed. 
More extensive experiments are much to be desired. 

The method of balancing slide valves may be divided into 
three classes : 

a. Removal of pressure from the back of the valve. 

b. Opposing the pressure on the valve by counter-pressure. 

c. Equalization of pressure on all sides. 

Typical examples of these three systems will here be given : 

a. Removal of pressure from back of valve. 

I. The so-called long Z^-valve, invented by Murray, and used 
on engines built by Watt, the pressure was relieved from the 
back by a form of stuffing box, which answered well, but was 
not adapted for high pressures. 

b. 





Fig. 1 1 66. 

2. Fig. Il66a shows the balance ring of Eoulton & Watt. 
The under side of the steam chest lid is finished parallel to the 
valve face. Against this surface a ring of soft cast iron or 
bronze is fitted steam tight, this ring being fitted to the valve 
by an elastic packing and moving back and forth with it. The 
space within the ring is subject only to the exhaust pressure. 
This form was used on the Great Eastern. 

Fig. ii66d shows the balance ring of Kirchweger, much used 
for locomotive engines. In this form the ring is pressed 
against the lid by steam pressure instead of spring packing. 
Both of these devices, as well as the similar ones of Penn, 
Borsig and others, leave too great a portion of the steam pres- 
sure unbalanced (at least 30 per cent, being left), and also pre- 
vent the valve from leaving its seat should water be carried 
into the cylinder X 

b. Balancing by counter-pressure. 

3. Cave's Valve with Balance Piston. 

The valve in this form. Fig. 1 167, is connected by a link to a 
piston, which works in a cylinder formed in the steam chest 
lid, and is subjected on the outside only to atmospheric 
pressure. Bourne's method of balancing is similar, except 
that the other side of the balance pisjon is in communica- 
tion with the exhaust. 

4. Valves with Rolling Support. Fig. 11 68. 

At a is shown I,indner's valve. The top of the valve 
itself is formed into a piston, sliding up and down in the 
valve and supported by two segmental rollers. 

J An elegant construction for this form of balanced valve is that of Robir/" 
son. (See Trans. Am. Soc. Mech. Engrs., Vol. IV, p. 375. 



286 



THE CONSTRUCTOR. 



The degree of balancing is dependent upon the size of the 
piston. At b is shown Armstrong's roller supported valve. In 
this case the valve is closed at top as usual, and the construc- 
tion is very simple and practical. At c is Bristol's valve, in 



(j 


^ \ 

'i' 


'1 


y 


^ 




1' 




SjWAV, >^,\ •i\ 


, iiiiii 1 




I 




4 


1 




IIV 



Fig. 1 167. 



which the valve is supported on a system of friction rollers. 
This has been used by the works at Seraing for large marine 
engines. To this class of methods of balancing belongs also 
that used by Worthington, in which a cylinder is formed in the 
top of the valve, as shown in Fig. 1016. 




Fig. II 68. 

5. Cuvelier's Valve, with pressure beneath,* Fig. 1169a. The 
ordinary slide valve is here combined with another, both being 
made in one piece, and the combined valve held down to its 
seat by pressure rollers. Live steam is admitted through the 
passage / into the space between the tv/o valves. At b is 
Fitch's valve, also with pressure beneath. In this form the 




Fig. 1 1 69. 

pressure rollers are omitted, and the valve is held down by live 
steam pressure in the steam chest. The steam is admitted 
through very small holes £ B, and escapes to the exhaust 
through similar holes B' B', so that the supply is about equal 
to the loss by condensation. An objection to the use of valves 
with pressure beneath is the large area of valve seat which is 
required. 




Fig. 1770. 

6. Double Seated Valves, Fig. 11 70. At a is Brandau's valve, 
and at b is Schaltenbrand's valve ; the former is analogous to 
Horublower's lift valve, Fig. Ii45ff, and the latter to the bell 



valve of Gros, Fig. 1 145^. In neither form is the degree of bal- 
ancing so complete as is desirable. 
c. Equalization of Pressure on all Sides. 




Fig. 1 171. 

7. A very complete equalization of pressure is obtained by 
making the valve in the form of a piston. Fig. 1171 shows a 
recently designed piston valve with its steam cylinder. The 
flat valve seat here becomes a cylinder, and the valve a double 
piston, the flat sides of the valve disappearing. The valve pis- 
tons are each fitted with a single ring behind which steam is 
admitted through a small hole, thus rendering springs unneces- 
sary. The principal defect in piston valves is the question of 
wear. The best results appear to be obtained by making the 
piston valve solid, and very accurately turned and polished, and 
made about yjjj" smaller in diameter than the boreof the valve 
cylinder, both valve and valve cylinder being made of the same 
material. Piston valves fitted in this manner last a long time. 

S. Rotary Valves : For steam hammers, in which valve gear 
operated by hand has been found preferable to automatic 
movements, valves with rotary movement, formed like cocks, 
are used to advantage. These have been well designed by 
Wilson, the superintendent of Nasmyth's works. 

a. h. 




Fig. 1172. 

Fig. 1172a shows an oscillating valve by Wilson. Opposite 
to the ports //, ///, II', are false ports or recesses of shallow 
depth. The steam enters at the end of the valve into the sym- 
metrical spaces /, /. The unbalanced area of the steam of the 
valve causes a corresponding endlong pressure which is received 
by a thrust bearing. If we neglect the slight pressure due to 
the steam in the false ports when expansion takes place in // 
and ///, the valve is balanced on all sides. Very large oscil- 
lating valves of this sort are easily moved by hand.f 

A modification of this valve enables it to be operated by rota- 
tion instead of oscillation, as shown in Fig. 1172^. Here the 
parts are symmetrically arranged, as was not the case with the 
old four way cock of Fig. 9S7. The exhaust passages IV con- 
nect with one end of the valve, and the admission /, with the 
other end. There remains here also an unbalanced end-long 
pressure which is received by a thrust bearing. With this ex- 
ception the valve is entirely balanced, and when well made the 
thrust bearing offers but little resistance. The construction of 
such valves demands a high degree of accuracj-, and a specialty 
of this form is made by the establishments of Dingier of Zwei- 
briicken and of Pfaffin Vienna. 

The brief examination which we have given to the preceding 
methods of balancing does not include a method which, while 
offering great difSculties of construction, appears to be gradu- 
ally coming into use. This method consists in surrounding the 
ordinary flat side valve with equalizing pressure plates. Several 
practical illustrations of this method will be given. 



* Compare Fig. ir34. 



tSee Zeitschr. D. Ing., i36S, Vol. II, p. 207. 



THE CONSTRUCTOR. 



287 



9. Wilson's Balanced Valve. Fig. 1 173. (First shown at the 
London Exhibition of 1862.) 




The valve is symmetrical, and slides between two parallel 
and similar faces, the lower face having opennigs correspond- 
ing to the ports, the upper face having similar false ports. The 
close fitting and accurate parallelism of the surfaces was de 
pended upon to obtain the balancing. In practice it was found 
that the balance i)lates would spring under the pressure of the 
steam unless made very stiff and strong, and that the weight of 
the valve caused much friction and wear. 

Both of these difficulties have been met in more recent 
designs, as will be seen below. 

a. b 




Fig. 1 174. 

10. Fig. It74n; shows the balancing, of the valves of the Por- 
ter-Allen engine.* The pressure plate is made very deep and 
stiff and formed with inclined plane bearings and set screws, by 
which the pressure can be very closely regulated. 

Fig. 1 1746 is Sweet's balanced valve. The pressure plate is 
here also made heavy and stiff, and is supported on longitu- 
dinal wedge bearings on each side, adjustable at the ends by 
screws. In both forms the pressure plate is fitted with springs 
to allow the plate to yield in case of water getting in the cylin- 
der. These forms of balanced valve have the objection that 
the ignorant mechanic may render the balancing ineffective by 
improper adjustment of the screws, permitting the full pressure 
of the live steam to act upon the valve. 

? 377-. 

Fl,UID V.\LVES. 

Valves may be formed of fluids, or, more generally speaking, 
may be constructed of pressure organs. Ratchets adapted to 
pressure organ?, as fluid valves are properly called, are in ex- 
tensive use, but have not generally been recognized as valves. 

a. b. 



^^r^aj p^^ 



^^^a^ ili'i 




Pig. 1175. 

They are all reducible to one of two principal forms, either the 
direct or inverted siphon. Fig. Ii75<z and h (compare ?. 312). 

A direct siphon connects two quantities of tlae same fluid 
above the level of both portions, these levels differing, for ex- 
ample by a height h ; an inverted siphon is similar, but con- 
nects them below the surface levels. Let (7, and a„ be liquids, 



which do not combine with a.-\ If the pressures of (7, and a., are 
equal, the fluid a will flow from the higher to the lower level 
under a pressure due to the height /;. In the inverted siphon 
the flow is constant, but with the direct siphon the flow is 
stopped, and the .siphon empties as soon as the level falls below 
the short end of the siphon. % If the upper vessel is again 
filled, the flow will begin as soon as the fluid attains the height 
//'of the bend in the siphon. The fluid in the siphon there- 
fore forms a valve, which converts a continuous flow into the 
upper vessel into a periodical flow into the lower one (see the 
example in I 324, where a similar action takes place with a 
rigid valve). This action of the siphon has recently been ap- 
plied (o excellent advantage. 

When the pressures in a^ and a.^ are different, as represented 
by the heights //, and //.^, as is frequently the case, we have for 
both forms for the height to which the flow is due, //i -{■ h — /;.^ 
for the height in an equivalent column of the fluid a. 

If this valve is positive, there will be an outflow, if it is zero, 
the fluid will be stalionarj', and if it is negative, there will be a 
reverse flow. In cases in which h^ -\- h — /;„ = O, h represents 
the measure of the difference between //.j and //,. It therefore 
follows that by means of fluid valves the relation between the 
fluids <7i and a, can be checked or controlled as may be desired. 

Applications of fluid valves are very numerous, as the fol- 
lowing examples will indicate : 




hil 



b. 



Ihj 



r^ 



■■^^ 



Fig. 1176a shows a 
water trap in a pipe. This 
is a fluid valve (inverted 
siphon) which checks a 
gas a, from mingling 
with a gas a^ so long as 
//, — /;.2 is less than twice 
the height s of the 
branches of the siphon. 
If the pressure from above 
upon a increases the over- 
flow runs off through a.,. 
This latter pipe must not 
be too small, however, or 
a siphon action will oc- 
cur, and all the water 
will be drawn off. This 
device is much used in 
gas works, chemical 
works, laboratories, etc. 

Fig. Ii76(^ shows the 
same arrangement used 
as a barometer, mano- 
meter, vacuum gauge, 
etc., the difference of 
level indicating differ- 
ences of pressure /;, — /;; 
for valves below 2 i. Applications of this principle are very 
numerous, from the largest forms to the most delicate physical 
instruments. 

Fig. 1177a is an open stand-pipe, used on certain forms of 
low pressure boilers. This is practically an inverted siphon, of 
which the boiler shell forms one branch. The fluid valve 
checks the steam a.^ against the atmosphere a^. If the pressure 
becomes so great that //.^ > /;, + '''' the fluid valve will be 
thrown out at the top of the pipe, the arrangement thus form- 
ing a safety valve against an excess of pressure in a.,. This 
device was for a long time in use for low pressure boilers, 
Brindley's feeding device, Fig. 1000, being constructed on this 
principle. Natural stand-pipes with periodical discharge exist 
as geysers. 

Fig. 11771^ is a closed stand pipe for steam boilers. The pipe 
which has first been filled with steam gradually fills with water 
as the steam condenses. If the water level in a sinks below 
the end of the pipe the water runs out and live steam fills the 
pipe again. This action is utilized in safetj' devices by Black 
and Warner, and by Schwartzkopf. 

In the blast furnace the fluid iron with the slag floating upon 
it forms an inverted siphon which checks the blast. In the 
Bessemer converter the air pressure is so great that the iron is 
kept in agitation by the air bubbling through it. 



Fig. II 76. 



* See Trans. Am. Soc. Mech. Engrs , Vol. IV, p. 26 
anced Valves. 



C. C. Collins, Bal- 



\1\\ this statement is included such fluids as do not :ningle by simple 
contact. In this sense steam and water will not mingle, and it thev are not 
of the same temperature the warmer will be transferred to the other. Air 
and water will not mingle because the w-aterhas become saturated with air- 
According to the researches of Colladon & Sturm (Memoire snr la compres- 
sion des liquides, 1827, reprinted by Schuchart, Geneva, 1S87), the saturation 
of water with air appears to partake of the nature of an internal, chemical 
combination. As m'ght be expected, water which is saturated with air 
shows a smaller compressibility in the Piezometer than water which is free 
from air, being 4S.65 millionths to 40.65 millionths. The combination of air 
with water ceases upon heating to the boiling point. 

X Natural inverted siphons with branches of varying levels exist in the 
case of artesian wells. 



288 



THE CONSTRUCTOR. 



In gas holders the water in the tank forming the seal is a 
fluid valve of tne inverted siphon type (compare Fig. 9486), 

a. b. 





Fig. 1 1 77. 

and a similar device is used with sand instead of water in Hoff- 
man's furnace, Fig. 1178, in which a^ is air, and a, smoke, the 
the bell-shaped lid being sealed with an annular valve of sand. 




F;g. ii;8. 

Fig. 1 179 is Wilson's water gas farnace.* In this a mixture of 
waste-slack and water forms a fluid valve. The mixture is 
propelled by an endless screw and discharged at the end. The 
atmosphere is at a^ and the gas at o^. the latter being kept 
under pressure bv a stesm jet. 




Fig. 1179. 

Hero's Fountain, Fig. 11 80, consists of two inverted siphon 
valves, in which a^ and «, have air at the atmospheric pressure, 
a^ is air under pressure, and a is water (often Cologne water). 
The action continues until the column /;', = h„. 



A practical application of the principle of Hero's fountain is 
the water trap of Morrison, Ingram & Co., Fig. iiSi.f In this 
device there is a periodical action of fluid valves as follows : a 
stream of water flows into the tank F aX E, gradually filling it, 
Fig. iiSia. The inner tube C, and fixed bell D, form an in- 
verted siphon, the shorter branch of which is the space between 




FiG. II 80. 

(Tand D. As soon as the level of the water in the tank jP rises 
above the top of Can overflow begins, filling the cup £, at the 
foot of the pipe C, and forming there a second siphon and 
making a seal between ^3 and a^, Fig. iiSiA. The two siphons 
now form a Hero's fountain, in which the continuing flow at E 

^ b. 

.1' i 




Fig. 1181. 



causes an outflow iuto the discharge pipe A. As the level con- 
tinues to rise in F, the air in a., becomes more and more com- 
pressed, until finally the pressure column h becomes greater 
than the difference in level of the lower siphon, causing its dis- 
charge and cousequent opening of the fluid valve into a^. This 
relieves the pressure on the air in a.,, thus permitting the upper 
siphon to act, and causing an immediate and rapid discharge 
of the contents of F. By adjusting the rate of flow at E this 
action can be regulated so as to take place periodically at any 
desired intervals of time. 

Richard's manometer. Fig. 11 82, consists of alternate direct 
and inverted siphons ; a is quicksilver, a, steam, a^ water and 
fij atmospheric air. 

The spiral pump and the Cagniardelle shown in Fig. 966a 
and b contain successive fluid valves in the same pipe, alter- 
nately direct and inverted. 

lyangen's device for discharging bone furnaces of the hot 
granular burnt bone, is a ratchet system involving valves con- 



*See A. Wilson,^Generation of Heating Gas, etc., Journal of Soc. of Chem- 
ical Industries, Manchester, Nov., 1883. 



t See Revue Industrielle, June, iS 



THE CONSTRUCTOR. 



289 



sisting of a granular pressure organ, Fig. H83. The discharge 
pipe d of the furnace is closed at the bottom by the sliding 
plate t: which is given a reciprocating movement (in this in- 
stance operated by a small hydraulic motor). This plate c is 
made with a step as shown in the figure at a, which receives a 
layer of the material, and on the return stroke, as shown at b. 




Fig. 1 182. 

this layer is discharged on the plate. This layer forms a suc- 
tion valve when acting as at a, and a discharge valve, as at b, 
while the plate c corresponds to a single acting piston, con- 
sidering the whole as a pump. If the plate c is made with 
a middle rib, as shown in Fig. iigoi;, it works both ways and 

b. c. 





Fig. II 83. 

becomes a double-acting pump. This is an illustration of the 
fluid valve in its most general form as applied to a pump. 

In many instances fluid valves are as good and sometimes 
even better than valves composed of rigid materials. Especially 
is this the case when they act continuously in one direction in 
in a free, open pipe, for which purpose they excel all other 
forms of valves, as in jet pumps and the like (see Fig. 972). 

S378. 
St.^tionary Valves. 

We have thus far considered valves as ratchets for pressure 
organs, when they operate so as to check the motion of the 
fluid at the intervals of time (see \ 365). If we consider this 
definition in its most ijeneral sense we may take it to include 
certain kinds of fastenings for closing apertures, and call these 
also valves. These we may distinguish from ordinary valves 
by the fact that they are not operated by the motion of the 
machine, and hence to them may be given the name of 
"stationary valves."' 

Stationary lift valves are found in the lids of steam cylinders, 
these belonging to the class of disk valves. These are required 
to resist internal pressure, and must therefore be securely 
bolted in place, the pressure being generally great, and resisted 
by the bolts. Steam chest covers are generally rectangular, 
flat, stationary valves, and an example of a stationary flap 
valve is seen in the valve chest door shown in Fig. 1128, this 
also being secured by means of bolts. Furnace doors, such as 
shown in Fig. 763, also belong to this class. The more readily 
such a valve is opened and closed the more nearly it approaches 
in construction to the movable valves, and packing is sometimes 
omitted in order to facilitate opening and closing. The valve 
chest lids, shown in Fig. 1131, are readily recognized, these 
being readily slipped into place and held by a yoke, or so- 
called "gallows screw." Numerous forms of stationary valves 
are also found in various kinds of bottle stoppers, these being 
effective substitutes for the older cork stoppers which often 
were held in place only by friction. Stationary fluid valves are 
also occasionally still found in use for bottle stoppers in parts 
of Italy and Greece. 




Fig. 11S4. 



In all the cases thus far mentioned the fastening by which 
the stationary valve is held in 
place must be at least slightly 
stronger than the pressure beneath 
the valve. 

As a stationary valve in which 
this is not the case, we have the 
ordinary manhole plate as used 
in steam boilers. Fig. 11 84. In _ 
this the pressure acts to hold the '^ 
plate to its seat. Other examples 
are found in the spring valves 
used in the so-called siphons of 
soda water, and the particular 
form of bottle stopper which con- 
sists of a small ball valve held up 
to the mouth of the bottle by the 
pressure within. Stationary 
slide valves are less frequently 
used than lift valves, as the con- 
ditions are less favorable for 
proper packing, but examples are 
to be found. It will be seen by 
the instances already given how 
far reaching into all buinches of 
machine design the use of ratchets 
for pressure organs extends. 

? 379- 
Stationary Machine Elements in General. 
It is not a peculiarit}' of valves alone to be used conveniently 
in the " stationary " form in the sense discussed in the preced- 
ing section. Here, as we have arrived at the close of the book, 
it is desirable to review the preceding pages in this respect. In 
the first four chapters of Section III the subjects considered are 
nearly always used as stationary elements. 

Rivets do not differ in form from cylindrical journals, but 
they are generally stationary because of two conditions ; be- 
cause of the firm binding of the surrounding metal, and because 
there are generally two or more rivets placed side by side. If 
only single rivet is used and no impediment to movement in- 
troduced, the binding of the metal would soon give way to any 
forces tending to cause rotation. 

Forced connections resemble journals and their bearings in 
form. The force by which the external piece grasps the inter- 
nal one eifectively resists all forces acting to produce rotation. 
Keyed connections are especially adapted for stationary service. 
The particular examples shown in Figs. 61S and 619 are in fact 
stationary keys in form, although really special cases of spiral 
gear wheels. Screws, in by far the greater number of cases, are 
used as stationary elements, probably in a greater variety of 
applications, broadly considered, than anj' other machine ele- 
ment. In ? 86 a glance is given at the use of the screw as an 
active machine element. 

Journals are frequently conveniently used as stationary ele- 
ments, as in the examples illustrated in Figs. 251, 252, 253, 256, 
257 and 258. In \ 90 we have already distinguished between 
"journals at rest" and "running journals," the former corres- 
ponding to the definition of stationary elements. Roller bear- 
ings for bridge truss supports, 'i 19S, are also stationary ele- 
ments. 

Crank connections are found in the bottle stoppers already 
mentioned, and in numerous other applications such connec- 
tions are properly considered as stationary elements, Here 
wheels are rarely used as stationary elements, but such applica- 
tions are frequently found of ratchet wheels. Longitudinal 
keys used to secure hubs upon their axles are almost invariably 
stationary elements, practically corresponding to "stationary 
ratchets," as a comparison between Figs. 1S8 and 654 will show. 
Ratchets also find numerous applications in stationary mechan- 
ism for securing bolts, keys and the like. An examination of 
Figs. 237 to 243 and 246 to 248 will illustrate this point. In the 
couplings shown in Figs. 423 to 430 we also have a number of 
stationary ratchets (see also Fig. 67S). 

In § 309 I have referred to the possibility of using pressure 
organs as standing or "stationary " elements, but these are as yet 
unimportant. The pipes used as conductors for pressure organs, 
however, furnish numerous instances of pressure organs. 

The above distinctions are by no means merel}^ theoretical, 
but are of a highly practical nature. Every means which will 
enable us to obtain a clearer and better comprehension of the 
use of machine elements should be most welcome. 

In the preceding arrangement the stationary elements have 
therefore been grouped together for this end. It follows that 
those forms which as " stationary " or "passive" elements are 
extensively used ia building and civil works, as well as in ma- 
chine design, forming the connecting links between the works 
of the civil and the mechanical engineer. 



THE CONSTRUCTOR. 



291, 



SECTION IV. 



MATHEMATICAL TABLES. 



?38o. 
Tables of Curves, Areas and Volumes. 

The following tables give in convenient form the most im- 
portant geometrical and mechanical properties of the more 
useful curves, areas and volumes. The significance of the let- 
ters used in the formulae will be found indicated on the dia- 
grams. The following remarks are also to be noted. 

By the rectification of a curve is meant the length j of that 
portion of the curve from the origin to the point x y, corres- 
ponding to the angle ^ ; and by S is meant the entire length of 
the curve. 

In the moment of inertia the mass of the body is assumed 
= I, in order to reduce the number of letters. In view of the 
importance of this subject a few points are here given. The mo- 
ments of inertia for surfaces are both equatorial and polar, each 
referred an axis of moments. This latter is called an equatorial 
axis when it lies in the plane of the surface, and a polar axis 
when it is at right angles to the surface. Each equatorial 
which passes through the centre of gravity is especially termed 
an equator-axis, and a polar axis which passes through the cen- 
tre of gravity is called a pole axis. Every surface, therefore, 
has but one pole-axis, and an infinite number of equator axes. 
The moment of inertia is called equatorial or polar, according 
to the axis to which it is referred. 

The moment of inertia Jp for any surface referred to the 
polar axis is found by adding together the two equatorial mo- 
ments of inertia Jg^ and Jf,,, the axes of which intersect each 
other at right angles in the polar axis : 



from the moment of inertia /referred to a parallel axis through 
S, by the following relation : 



Jp= Jq\ + Jt: 



(4.6) 



J'= J-\-d'- F. 



(417) 



The moment of inertia /' of a surface, referred to any axis 
situated at a distance a, from the centre of gravity 5, is found 



in which j 'is the area of the surface. This relation also holds 
good for solids, if the mass of the body is substituted for F. 

For solids one of the preceding conditions does not hold. 
For each different shape one of the axes which passes through 
the centre of gravity, is taken as the pole-axis for all sections 
normal to it, and the section at right angles to this axis which 
passes through the centre of gravity is called the Equatorial 
Section, whence the equatorial and polar moments of inertia 
are in these cases distinguished according to their position with 
regard to this equatorial section. In all the examples of solids 
here given, the actual equatorial and polar axes are meant. 

For a right prism, of any given base having as the polar mo- 
ment of inertia ip and the half-height = /, the polar moment 
of inertia is : 

Jp = ^-lip {418) 

and the moment of inertia referred to an equatorial axis : 

Jp=VifP-\-2lig (419) 

in which y" is the area of the cross section, and i,/ the equatorial 
moment of inertia of the cross section referred to the same 
axis a.^Jg. 

The centre of gravity and the moment of inertia for a surface 
of irregular form is often readily obtained by grapho-static 
methods, with sufBcient numerical accuracy. For this purpose 
the force and cord polygons are applicable according to the 
methods already described in Section II. 



292 



THE CONSTRUCTOR. 



« 



M 
A 
P 

< 
> 

P 

o 

h 
o 

to 
P 


<l 



0, i/^ 




'ni^ "I' 



H Ms 



+ 



+ 






+ 



Oh 

g 

O 



o 



ts -o 



o 




^1"^ 



o 

M 

< 

P 

w 

< 
o 



o I 
s + 



^ II 

o 



pH^i Ph 



+ 



g 
o 

Ph 



+ 



.+ 



•«^ 



I ^ 



l| ^ 



Pi 

H 

O 
P4 

Ph 

< 
u 
H 



o 

< 
p 
01 

W 

a 
< 
a 
p 
o 
2; 
<i 

o 

w 



> 

Pi 
P 
O 



o 



a; 

.11 






+ 



1 


II 


+ 


>^ 


^ 

^ 


\ 


6 


t^ 


^ 


s 






a 




P^ iu 







ft 

a 



t*. 


.i^^i 


0^ 


M y 


II 


11 h 


Co 


^ II 


'^ 


^^ 



^ 



^r 



ii I 



.^ ^ 



3 

z 



^ 






Si « 



^1^ I I T 
II \i J 

% c ll 



a 






H 

ll 




+ 







^, 




\ 


\ ^ 


1 .) 


VH 


17 


\^ 


7 



A 


w 


Oi 




^ 


^v- 


// 


> 


\\.a 


f 


/ 




V 




\ 


// 


/ i" 




\ 



i+r ^ 



> 



Q 
II 
<1 



a 

o 

6^ 




THE CONSTRUCTOR. 



293 



A 
< 

W 



w 

A 

H 

> 

U 
O 



3 

S 
< 



^ g 

a « 

a I 

ho "S 



11 



A .« .2 c 



p. Cc; 






O 13 



ii O 



S ^ 



o 



o 
'0 



0. aJ ^ -5 - 



c; .^ a 



cti rf ^ o *^ 



o 



o r^ -2 



ll \1 



CJ " •— ' 






>^ ^ 





a 
8 

CD 
^0 




s 








a 


0) 





'tT 





,n 


> 


OJ 


10 


>> 




















en 


u 




II 


M 




OJ 





1^ 


a 


+-< 


C:: 


bf 


X 





^-1 



II 




CO 

II 

1^ 















^ 







p 


<4-< 








fTl 






























nd 


II 









a; 



















■^ 


=b 


OJ 







ft? 

I? 



il.t 



3 I CN 



Ml 



ill 



* 

^ 



o 



rCrft?: 



O Q 



5 a 



3 ° 
a 



"mm 

a a a 
.2.2-2 

a 3£ 

> > ^ 

O) OJ OJ 



ft? . 



ft, 



bo 

a 



S 
ft. 



^ 



+ 



> 



^ N 




o 



pa 






taOCX2 



294 



THE CONSTRUCTOR. 




Moment of Inertia. 



For polar axis through C: 

■''- ? 4 

For Pole axis through centre of 
gravity 5 : 

>-*/^ I — C05/3' 



r 1^ .,1 I COS 



-cosp\ r*[ i—cosf}^ 



For polar axis through C : 

'"~ 2 ' ~ ^ ' 
For polar axis through S : 

i 
r I 2 \ 

For the equatorial axes XX, Y Y : 

ji f'- 

4 



^-=-^' = ^ = T-- 



For Pole axis through C: 

'- 1 2 

For Equator axis A' A": 



For polar axis through C : 



= f('V-'V)=^(4*H+r*3) 



For polar axis through C : 
' 4 4 ^- 12 J 



•> 



- 2 siti ft cos^ ji COS /S sin' /3 1 



For eauatorial axes A' A' and Y Y 



Jx 



y 



5 15 



35 loi 



For Equator axis A' A':/ ,= — a b^ 

4 
For Pole axis C : 



Jx 

Jy=, 



6 ~ ~i2~'-^-"~'"Ts~~^6" 



6 

/;2 b h 



bh^ 
bli' h, , ., bli 



(«3 4- ifi) 



/5=— +-^(«'+-.'»)--[8 («-^+Z'-')-3 b--^ 



THE CONSTRUCTOR. 



295 



No. 



Form. 



XXI. 

Triamlar 
Prisi. 



XXII. 

Rectangnlar 
Frisi. 



XXIII. 



RloiMc Prism, 



XXIV. 

Hexaional 
Prisi. 



XXV. 

Cylinder. 



XXVI. 

Hollow 
CyMer. 



XXVII. 

ParaMic Prisi, 




Sides : F^ = 2l (a-\- b + c] 
One end : P„ = 








Q. ^nt 




.-*-Q 





Surface. 



Volume. 



V= b h I 



Centre of 
Gravity. 



Moment of Inertia. 



For Equator axis Q Q : 



7. 






Centre of 
Figure. 



Sides: f,=4/ {b + /i) 
One end : F„^ b h 



v; 



Sides :/=",= 8/ V/;" + 



One end : K,= b h 



For Pole axis P F : 



Ip = '« |_~i~ + "c: (" + ^) — 



For Equator axis Q O 



Jc 



V=2bhl 



4' r=2bhl 



Sides : /^, = 1 2 / r 

One end : /^.^ ^-r'y/T,- 
= 2.sgSr' 



= "'(y+S) 



Centre of 
Figure. For Pole axis /'/': 



111 
12 



// = ^(^'-^ + *') 



Centre of 
Figure. 



= 5.196 / r- 



= 1 Centre of 



) 



For Eqiiator axis O O : 
For Pole axis P P\ 



For Equator axes Q Q and O^ Q^ : 



/, 



'^='"(t 



+ - 



24 



Vertical surface : 
F, = A^lr 

One end : F.^= tt i^ 



Figure. I For Pole axis /'P: 



r=2^/;'=' 



Centre of 
Figure. 



-// = !; "--^ 



For Equator axis O Q : 



7y = '" 



For Pole axis P P: 



Vertical surface : I 

One end : ^ 

^2--^"(''i' — ''■/) =2-rb 



I 



For Equator axis O Q ■ 



Centre of j 
Figure. ! 



r /'^ r'^ ^^ 
L3 2 ^ S 



] 



For Pole axis P P: 



jp =-- "' ['V-' + '•;-■] = '« Q-'+ -^]] 



One end : F.^^ — jc y 
3 



r= — I xy 
3 



For Equator axis O O : 



^-'"[M'] 



For Pole axis P P: 



Jp = '« 



L 5 35 J 



296 



THE CONSTRUCTOR. 



No. 



XXVIII. 



XXIX. 

Eectanplar 
Pyraiid. 



XXX. 

RilM CoEe. 



XXXI. 

Triicateil Cone, 



Form. 





XXXII. 

SDiere. 



XXXIII. 

Sector of spliere, 



XXXIV. 

sesmem of 
Spliere. 



XXXV. 



SDtieroil. 



XXXVI. 

Paralioloid 01 
RevoMion. 







Surface. 



F=\^-'-Rr 



Sides : 



v; 



4 

4 
Bottom : F„^ a b. 



luclined surface: 

i^j ^ 77 JV h- 4- r- = TT >-5 

Bottom : F„^=ti r''- 



Sides ; 
F,^~ [r, + r.^VlfiI^^^^,^^ 

= 2 TV }■ S 

Ends : 



F^^r'Tz 



Conical surface ; 
F^= a~ r=~ r v 2 ;' /; — K' 



Curved surface : 





Bottom : 



F2 = a' ;r, r 



2 h 



Bottom: F^^t^ y' 



Volume. 



V^2 n- J?r 



^ h a b h 
3 3 



~ 3 



r=-/;[r,''+r,;-,+r/] 



7'= — TT r'^ h 
3 



Centre OP Gravity Moment of Inertia. 



Centre of Figure. 



Jt 



^~ 4 
For the surface onlj- 
. h 



4\ ri'+ri^+r,- 



Centre of Fi^rure. 



-10-4) 



J'=n/i' I r 



■ /^ (3 «» + /'') 



4 ir~/i 
For the surface only 
. /i 



V= — TV a be 



2 



Centre of Figure. 



For Equator axis Q Q : 



Jo = "' 



2^8 



For Pole axis P P 
3 . 



R'-\-- 






For Equator axis Q Q : 



Jc 



:r3./,+ i!-i 

Lso ^ 20 J 



? " I So ■ 20 . 

For Pole axis P P : 
117 



^p- 



For Equator axis Q Q : 



Jo = — "' 
y 20 



D-f] 



For Pole axis P P : 

Jh = - - "' '^ 
/^ 10 



For Pole axis P P: 



Jt, 



3 i\ — » 2 
10 ;■/ — ?v 



For Equator axis O Q : 



For Pole axis P P : 



For Pole axis P P : 



+ 



20 J ir — h 



For Equator axis Q Q, 
coincident with a : 



For Equator axis Q Q : 

/.="' (f +S) 

For Pole axis P P: 
in 



Jp- 



-r 



THE CONSTRUCTOR. 



297 



381 



Trigonometrical Table. 

The following table contains, in convenient form, the sines, cosines, tangents and cotangents for angles from 0° to 90° 
lor every ten minutes, and also the corresponding arcs to a radius of unity. At the foot of the table arcs are also given for 
small angles and also for some of the more frequently used angles greater than 90°. 



ANGLE. 












" 


ANGLE. 


1 
ANGLE. 














ANGLE. 




arc. 


sine, cosine. 


tan. 


cot. 


arc. 






arc. 


sine. 


cosine. 


tan. 


cot. 


arc. 




deg. inin, 






deg. 


min. 


deg. 


min. 


deg. 


min. 








0.0000 


0.0000 


1. 0000 


0.0000 


CO 


1.5708 


90 





10 





0.1745 


0.1736 


0.9848 


0.1763 


5-6713 


1-3963 


80 







10 


0.0029 


0.0029 


1. 0000 


0.0029 


343.77 


1.5679 




50 




10 


0.1774 


0.1765 


0.9843 


0.1793 


5-5764 


1-3934 




50 




20 


0.005S 


0.0058 


1. 0000 0.0058 


171.89 


1.5650 




40 




20 


0.1804 


0.179+ 


0.9838 


0.1823 


5-4S45 


1-3904 




40 




30 


0.0087 


0.0087 


1. 0000 


0.0087 


114.59 


1. 5621 




30 




30 


0.1833 


0.1822 


0.9833 


0.1853 


5-3955 


1-3875 




30 




40 


0.0116 


0.0116 ' 0.9999 


0.0116 


85.940 


1.5592 




20 




40 


0.1862 


0.1851 


0.9827 


0.1S83 


5-3093 


1.3S46 




20 




50 


0.0145 0.0145 1 0.9999 

( 1 


0.0145 


68.750 


1.5563 




10 




50 


0.1891 


0.1880 


0.9S22 


0.1914 


5-2257 


1.3817 




10 


I 


' 1 
0.0175 0-OI75 0.9998 0.0175 


57.290 


1-5533 


89 





II 





0.1920 


0.190S 


0.9816 


0.1944 


5-1446 


1.3788 


79 







10 0.0204 ■ 0.0204 [ 0.9998 0.0204 


49.104 


1.5504 




50 




10 


0.1949 


0.1937 


0.9811 


0.1974 


5-0658 


1-3759 




50 




20 0.0233 ' 0.0233 : 0.9997 0.0233 


42.964 


1.5475 




40 




20 


0.197S 


0.1965 


0.9805 


0.2004 


4.9894 


1-3730 




40 




30 0.0262 


0.0262 0.9997 , 0.0262 


38.188 


1.5446 




30 




30 


0.2007 


0.1994 


0.9799 


0.2035 


4.9152 


1-3701 




30 




40 0.0291 


0.0291 


0.9996 1 0.0291 


34.36S 


I-5417 




20 




40 


0.2036 


0.2022 


0.9793 


0.2065 


4.8430 


1-3672 




20 




50 


0.0320 


0.0320 


0.9995 0.0320 


31.242 


I-538S 




10 




50 


0.2065 


0.2051 0.9787 


0.2095 


4.7729 


1.3643 




10 


2 





0.0349 


0.0349 


0.9994 0.0349 


28. 636 


1.5359 


88 





12 





0.2094 


0.2079 0.97S1 


0.2126 


4.7946 


1.3614 


78 







10 


0.0378 


0.0378 1 0.9993 ; 0.0378 


26.432 


1.5330 




50 




10 


0.2123 


0.2108 1 0.9775 


0.2156 


4.6382 


1.3584 




50 




20 


0.0407 0.0407 1 0.9992 1 0.0407 


24.542 


1. 5301 




40 




20 


0.2153 


0.2136 0.9769 


0.2186 


4.5736 


1-3555 




40 




30 


0.0436 0.0436 0.9990 0.0437 


22.904 


1.5271 




30 




30 


0.2182 


0.2164 


0.9763 


0.2217 


4.5107 


1.3526 




30 




40 


0.0465 


0.0465 0.9989 0.0466 


21.470 


1.5243 




20 




40 


0.2211 


0.2193 


0.9757 0.2247 


4.4494 


1-3497 




20 




50 


0.0495 


0.0494 0.9988 0.0495 


20.206 


1.5213 




10 




50 


0.2240 


0.2221 


0.9750 0.2278 


4.3897 


1.3468 




10 


3 





0.0524 


0.0523 0.9986 0.0524 


19.081 


1.51S4 


87 





13 





0.2269 


0.2250 


0.9744 0.2309 


4.3315 


1-3439 


77 







10 


0.0553 


0.0552 


0.9985 1 0.0553 


18.075 


1.5155 




50 




10 


0.229S 


0.2278 0.9737 


0.2339 


4.2747 


1.3410 




5° 




20 


0.0582 


0.0581 


0.9983 0.0582 


17.169 


1.5126 




40 




20 


0.2327 


0.2306 


0.9730 


0.2370 


4.2193 


1-3381 




40 




30 


0.061 1 


0.0610 


0.9981 0.0612 


16.350 


1.5097 




30 




30 


0.2356 0.2334 


0.9724 


0.2401 


4.1653 


1-3352 




30 




40 


0.0640 


0.0640 


0.9980 j 0.0641 


15.605 


1 .5068 




20 




40 


0.2385 


0.2363 


0.9717 


0.2432 


4.1126 


1-3323 




20 




50 


0.0669 


0.0669 


0.9978 


0.0670 


14.924 


1.5039 




10 




50 0.2414 


0.2391 


0.9710 


0.2462 


4.0611 


1-3294 




10 


4 


0.0698 1 0.069S 


0.9976 


0.0699 


14.301 


1.5010 


86 





H 





0.2443 


0.2419 


0.9703 


0.2493 


4.0108 


1.3264 


76 





10 


0.0727 ' 0.0727 


0.9974 ; 0.0729 


13.727 


1.49S1 




50 




10 


0.2473 


0.2447 


0.9696 


0.2524 


3.9617 


1-3235 




50 


20 


0.0756 : 0.0756 


0.9971 0.075S 


13.197 


1.4951 




40 




20 


0.2502 


0.2476 


0.9689 


0.255s 


3-9136 


1.3206 




40 




30 


0.0785 0.07S5 0.9969 


0.0787 


12.706 


1.4923 




30 




30 


0.2531 


0.2504 


0.9681 ' 0.2586 


3.S667 


1-3177 




30 




40 


0.0814 1 0.0814 


0.9967 


0.0S16 


12.251 


1 .4893 




20 




40 


0.2560 


0.2532 


0.9674 


0.2617 


3.8208 


1.3148 




20 




50 


0.0844 ' 0.0843 


0.9964 


0.0846 


11.826 


1 .4864 




10 




50 


0.2589 


0.2560 


0.9667 


0.264S 


3.7760 


1-3119 




10 


5 





0.0873 


0.0S72 


0.9962 0.0S75 


1 1 .430 


1.4835 


85 





15 





0.2618 


0.2588 


0.9659 


0.2679 


3.7321 


1.3090 


75' 







10 


0.0902 


0.0901 


0.99S9 0.0904 


11.059 1.4806 




50 




10 


0.2647 


0.2616 


0.9652 


0.2711 


3.6891 


1.3061 




50 




20 


0.0931 


0.0929 ; 0.9957 j 0.0934 


10.712 


1-4777 




40 




20 


0.2667 


0.2644 


0.9644 


0.2742 


3.6470 


1.3032 




40 




30 


0.0960 


0.0958 0.9954 ! 0.0963 


10.385 


1.4748 




30 


30 


0.2705 


0.2672 


0.9636 


0.2773 


3.6059 


1.3003 




30 




40 


0.0989 


0.0987 


0.9951 1 0.0992 


10.078 


1.4719 




20 


40 


0.2734 


0.2700 


0.9628 


0.2805 


3-5656 


1.2974 




20 




50 


0.1018 


0.1016 


0.9948 


0.1022 


9.7S82 


1 .4690 




10 




50 


0.2763 


0.2728 


0.9621 


0.2836 


3-5261 


1.2945 




10 


6 





0.1047 


0.1045 


0.9945 


0.1051 


9-5144 


1. 4661 


84 





16 





0.2793 


0.2756 


0.9613 \ 0.2867 


3-4874 


1. 2915 


74 







10 


0.1076 


0.1074 


0.9942 


0.1080 


9-2553 


1.4632 




50 


TO 


0.2822 ' 0.2784 


0.9605 ' 0.2S99 


3-4495 


1.28S6 




50 




20 


0.1 105 


0.1103 


0.9939 0.1 110 


9.009S 1.4603 


40 


20 1 O.2S5I O.2S12 


0.9596 0.2931 


3.4124 


1.2857 




4(1 




30 


0.1134 


0.1132 


0.9936 j 0.1139 


8.7769 


1.4573 




30 




30 


0.2880 0.2840 


0.9588 0.2962 


3-3759 


1.2828 




30 




40 j 0.1 164 


0.1161 


0.9932 0.1 169 


S.5555 


1-4544 




20 




40 


0.2909 0.2868 


0.9580 0.2994 


3-3402 


1.2799 




20 




50 0.1193 


0.1190 


0.9929 1 0.1198 

1 


8.3450 


1-4515 




10 




50 


0.2938 


0.2896 


0.9572 


0.3026 


3-3052 


1.2770 




10 


7 


0.1222 


0.1219 


0.9925 1 0.1228 


8.1443 


1 .4486 


83 





17 





0.2967 


0.2924 


0.9563 


0.3057 


3.2709 


1.2741 


73 







10 


0.1251 


0.1248 


0.9922 


0.1257 


7.9530 


1.4457 




50 




10 


0.2996 


0.2952 


0.9555 


0.3089 


3-2371 


1. 2712 




50 




20 


0.12S0 


0.1276 


0.991S 


0.1287 


7.7704 1.4428 




40 




20 


0.3025 0.2979 


0.9546 '0.3121 


3.2041 


1.2683 




40 




30 


0.1309 0.1305 


0.9914 0.1317 


7-5958 1,4399 




30 




30 


0.3054 


0.3007 


0.9537 


0.3153 


3.1716 


1.2654 




30 




40 


0.133S 0.1334 


0.9911 ' 0.1346 


7-4287 


1.4370 




20 




40 


0.3083 


0.3035 


0.9523 


0.31S5 


3.1397 


1.2625 




20 




50 0.1367 0.1363 0.9937 0.1376 


7.2687 


I. 434 I 




10 




50 


0.3113 


0.3062 


0.9520 


0.3217 


3.1084 


1.259s 




10 


8 


1 0.1396 0.1392 ; 0.9903 0.1405 


7.1154 


1-4312 


82 





18 





0.3142 


0.3090 


0.95 II 


0.3249 


3.0777 


1.2566 


72 







10 


0.1425 


0.1421 0.9899 1 0.1435 


6.9682 


1.4283 




50 




10 


0.3171 


0.3118 


0.9502 


0.3281 


3.0475 


1.2537 




SO 




20 


0.1454 


0.1449 0.9894 I 0.1465 


6.S269 


1.4254 




40 




20 


0.3200 


0.3145 


0.9492 


0.3314 


3.0178 


1.2508 




40 




30 


0.1484 


0.147S 1 0.9S90 


0.1495 


6.6912 


1.4224 




30 




30 


0.3229 


0.3173 


0.9483 


0.3346 


2.98S7 


1.2479 




30 




4^ 


0.1526 


0.1507 0.9S86 


0.1524 


6.5606 


1. 4195 




20 




40 


0.3258 


0.3201 1 0.9474 


0.3378 


2.9600 


1.2450 




20 




5^ 


0.1542 


0.1536 0.9S81 


0.1554 


6.4348 ' 1. 4166 

1 




10 




50 


0.3287 


0.3228 0.9465 


0.341 1 


2.9319 


i.2421 




10 


9 





0.1571 


0.1564 1 0.9877 


0.1584 


6.3138 


1-4137 


81 





19 





0.3316 


0.3256 


0.9455 


0.3443 


2.9042 


1.2392 


71 







10 


o.i6oo 


0.1593 0.9872 j 0.1614 


6.1970 


I. 4108 




50 




10 


0.3345 


0.3283 


0.9446 


0.3476 


2.8770 


1-2363 




50 




20 


0.1629 


0.1622 


C.986S 0.1644 


6.0844 


1.4079 




40 




20 


0.3374 


0-3311 


0.9436 


0.350S 


2.8502 


1-2334 




40 




33 


0.165S 


0.1650 


0.9863 ; 0.1673 


5.9758 


1.4050 




30 




30 


0.3403 i 0.333S 


0.9426 


0.3541 


2.S239 


1-2305 




30 




40 


0.1687 


0.1679 


O.985S 


0.1703 


5.870S 


1. 402 1 




20 




40 


0.3432 


0.3365 


0.9417 


0.3574 


2.79S0 


1.2275 




20 




50 


0.1716 


0.1 70S 


0.9853 


0.1733 


5-7694 


1.3992 




10 




SO 


0.3462 


0.3393 


0.9407 


0.3607 


2.7725 


1.2246 




10 


Angle. 


arc. 


cosine. 


sine. 


cot. 


tan. 


arc. 


Angle. 


Angle. 

1 


arc. 


cosine. 


sine. 


cot. 


tan. 


arc. 


Angle. 



298 



THE CONSTRUCTOR. 



ANGLE. 














ANGLE. ANGLE. 














ANGLE. 




arc. 


sine. 


cosine. 


tan. 


cot. 


arc. 




arc. 


sine. 


cosine. 


tan. 


cot. 


arc. 




deg. 


min. 


deg. 


min. 


Ideg. 


min 


deg. 

59 


rain. 


20 





0.3491 


0.3420 


0.9397 


U.3640 


2.7475 


1. 2217 


70 





31 





0.5411 


0.5150 


0.S572 


0.6009 


1.6643 


1.0297 







10 


0.3520 


0.3448 


0.9387 


0.3673 


2.7228 


1.2188 




50 




10 


0.5440 


0.5175 


0.8557 


0.604S 


1.6534 


1.0268 




50 




20 


0.3549 


0.3475 


0.9377 


0.3706 


2.6985 


1.2159 




40 




20 


0.5469 


0.5200 


0.8542 


0.6088 


1.6426 


1.0239 




40 




30 


0.3578 


0.3502 


0.9367 


0.3739 


2.6746 


1.2130 




30 




30 


0.5498 


0.5225 


0.8526 


0.6128 


1.6319 


1.0210 




30 




40 


0.3607 


0.3529 


0.9356 


0.3772 


2.6511 


1.2101 




20 




40 


0.5527 


0.5250 


0.8511 


0.616S 


1.6212 


1.0181 




20 




50 


0.3636 


0.3557 i 0.9346 ! 0.3S05 

1 1 


2.6279 


1.2072 




10 




50 


0.5556 


0.5275 ; 0.8496 

1 


0.6208 


1.6107 


1.0152 




10 


21 





0.3665 


0.3584 


0.9336 


0.3839 


2.6051 


1.2043 


69 





32 





0.5585 


0.5299 1 0.8480 


0.6249 


1.6003 


1 .0123 


S8 







10 


0.3694 


0.3611 


0.9325 


0.3872 


2.5826 


I. 2014 




50 




10 


0.5614 


0.5324 0.8465 


0.6289 


1.5900 


1 .0094 




SO 




20 


0.3723 


0.3638 


0.9315 


0.3906 


2.5605 


1.1985 




40 




20 


0.5643 


0.5348 0.8450 


0.6330 


1.5798 


1 .0065 




40 




30 


0.3752 


0.3665 1 0.9304 


0.3939 


2.5386 


1. 1955 




30 




30 


0.5672 


0.5373 ; 0.8434 


0.6371 


1.5697 


1.0036 




30 




40 


0.3782 


0.3692 


0.9293 


0.3973 


2.5172 


1.1926 




20 




40 


0.5701 


0.5398 


0.8418 


0.6412 


1.5597 


1.0007 




20 




50 


0.381 1 


0.3719 


0.9283 


0.4006 


2.4960 


1.1897 




10 




50 


0.5730 


0.5422 


0.8403 


0.6453 


1.5497 


0.9977 




10 


22 


[ 0.3840 


0.3746 


0.9272 


0.4040 


2.4751 


1.1868 


68 





33 





0.5760 


0.5446 


0.8387 


0.6494 


1.5399 


0.994S 


57 







10 


0.3869 


0.3773 


0.9261 


0.4074 


2.4545 


1.1839 




50 




10 


0.5787 


0.5471 


0.8371 


0.6536 


1.5301 


0.9919 




50 




20 


0.3S9S ' 0.3S00 


0.9250 


0.4108 


2.4342 


1.1810 




40 




20 


0.5818 


0.5495 


0.8355 


0.6577 


1.5204 


0.9890 




40 




30 


0.3927 0.3S27 0.9239 


0.4142 


2.4142 


1.1781 




30 




30 


0.5847 


0.5519 


0.8339 


0.6619 


1. 5108 


0.9861 




30 




40 


0.3956 j 0.3854 1 0.922S 


0.4176 


2.3945 


1.1752 




20 




40 


0.58-6 


0.5544 


0.8323 


0.6661 


1.5013 


0.9832 




20 




50 


0.39S5 


0.3S81 


0.9216 


0.4210 


2.3750 


1.1723 




10 




SO 


0.5905 


0.5568 0.8307 


0.6703 


I. 4919 


0.9803 




10 


23 





0.4014 


0.3907 


0.9205 


0.4245 


2.3559 


1. 1694 


67 





34 





0.5934 0.5592 0.8290 


0.6745 


1 .4826 


0.9774 


56 







10 


0.4043 


0.3934 


0.9194 


0.4279 


2.3369 


1.1664 




50 




10 


0.5963 


0.5616 


0.8274 


0.6787 


1.4733 


0.9745 




50 




20 


0.4072 


0.3961 


0.9182 


0.4314 


2.3183 


1.1636 




40 




20 


0.5992 


0.5640 


0.8258 


0.6830 


1.4641 


0.9716 




40 




30 


0.4102 


0.3987 


0.9171 


0.4348 


2.2998 


1. 1606 




30 




30 


0.6021 


0.5664 


0.8241 


0.6873 


1.4550 


0.9687 




30 




40 


0.4131 


0.4014 


0.9159 


0.4383 


2.2817 


1. 1577 




20 




40 


0.6050 


0.5688 


0.8225 


0.6916 


1.4460 


0.9657 




20 




50 


0.4160 


0.4041 


0.9147 


0.4417 


2.2637 


1.1548 




10 




50 


0.60S0 


0.5712 


0.8208 


0.6959 


1.4370 


0.9628 




10 


24 





0.4189 


0.4067 


0.9135 


0.4452 


2.2460 


1.1519 


66 





35 





0.6109 


0.5736 


0.8192 


0.7002 


1.4281 


0.9599 


55 







10 


0.4218 


0.4094 


0.9124 


0.4487 


2.22S6 


1.1490 




50 




10 


0.6138 


0.5760 


0.8175 


0. 7046 


I.4I73 


0.9570 




SO 




20 


0.4247 


0.4120 


0.91 12 


0.4522 


2.2113 


1.1461 




40 




20 


0167 


0.5783 


0.8158 


0.7089 


1. 4106 


0.9541 




40 




30 


0.4:^76 


0.4147 


0.9100 


0.4557 


2.1943 


1.1432 




30 




30 


0.6 96 


0.5807 


0.8141 


o.7'33 


I. 4019 


0.9512 




30 




40 


0.4305 


0.4173 


0.908S 


0.4592 


2.1775 


i.ii03 




20 




40 


0.0225 


0.5831 


0.8124 


0.7177 


1.3934 


0.9483 




20 




50 


0.4334 


0.4200 


0.9075 


0.4628 


2.1609 


'■1374 




10 




50 


0.6254 


0.5854 


0.8107 


0.7221 


1.3848 


0.9455 




10 


25 





0.4363 


0.4226 


0.9063 


0.4663 


2.1445 


1.1345 


65 





36 





0.6283 


0.5878 


0.8090 


0.7265 


1.3764 


0.9425 


54 







10 


0.4392 


0.4253 


0.9051 


0.4699 


2.1283 


1.13ID 




50 




10 


0.6312 


0.5901 


0.8073 


0.7310 


1.3680 


0.9306 




5° 




20 


0.4421 


0.4279 


0.9038 


0.4734 


2.1123 


1.1280 




40 




20 


0.6341 


0.5925 


0.S056 


0.7355 


1.3597 


0.9367 




40 




30 


0.4451 


0.4305 


0.9026 


0.4770 


2.0965 


1. 1257 


30 1 




30 


0.6370 


0.5948 


0.S039 


0.7400 


1.3514 


0.9338 




30 




40 


0.4480 


0.4331 


0.9013 


0.4806 


2.0S09 


I. 1228 




20 




40 


0. 6400 


0.5972 


0.8021 


0. 7445 


1.3432 


0.9308 




20 




50 


0.4509 


0.4358 


0.9001 


0.4841 


2.0655 


1.1199 




10 




50 


0.6429 0.5995 


0. 8004 


0. 7490 


1.3351 


0.9279 




10 


26 





0.453S 


0.4384 


0.8988 


0.4877 


2.0503 


I. 1170 


64 





37 





I 
0.6458 0.601S 


0.7086 


0.7536 


1.3270 


0.9250 


53 







10 


0.4567 


0.4410 


0.8975 


0.4913 


2.0353 


I. 1141 




50 




10 


0.6487 


0.6041 


0.7069 


0.7581 


1.3190 


0.9221 




50 




20 


0.4596 


0.4436 


0.S962 


0.4950 


2.0204 


1.1112 




40 




20 


0.6516 


0.6065 


0.7951 


0.7627 


1.3111 


0.9192 




40 




30 


0.4625 


0.4462 


0.8949 


0.4986 


2.0057 


I. 1082 




30 




30 


0.6545 


0.60S8 


0.7934 


0.7673 


1.3032 


0.9163 




30 




40 


0.465 A 


0.4488 


0.8936 


0.5022 


1.9912 


I. 1054 




20 




40 


0.6574 


0.6111 


0.7916 


0.7720 


1.2954 


o.9>34 




20 




50 


0.4683 


0.4514 


0.8923 


0.5059 


1.9768 


1.1025 




10 




50 


0.6603 


0.6134 


0.7S98 


0.7766 


1.2876 


0.9105 




10 


27 





0.4712 


0.4540 


0.8910 


0.5095 


1.9626 


1 .0996 


63 





38 





0.6632 


0.6157 


0.7S80 


0.7813 


1.2799 


0.9076 


52 







10 


0.4741 


0.4566 


0.8897 


0.5132 


1.9486 


1 .0966 




50 




10 


0.6661 


0.6180 


0.7862 


0.7S60 


1.2723 


0.9947 




50 




20 


0.4771 


0.4592 


0.S884 


0.5169 


1.9347 


1.0937 




40 




20 


0. 6690 


0.6202 


0.7844 


0.7907 


1.2647 


0.9018 




40 




30 


0.4800 


0.4617 


0.8870 


0.5206 


1.9210 


1 .0908 




30 




30 


0.6720 


0.6225 


0.7826 


0.7954 


1.2572 


0.8988 




30 




40 


0.4829 


0.4643 


0.8857 


0.5243 


1.9074 


1.0879 




20 




40 


0.6749 


0. 6248 


0.7S08 


0.8002 


1.2497 


0.S959 




20 




50 


0.4858 


0.4669 


0.8843 


0.5280 


1 .8940 


1 .0850 




10 




SO 


0.6778 


0.6271 


0.7790 


0.8050 


1.2423 


0.8930 




10 


28 





O.4S87 


0.4695 


0.S820 


0.5317 


1.8807 


1.0821 


62 





39 





0.6807 


0.6293 


0.7771 


0.S098 


'.2349 


0.8901 


51 







10 


0.4916 


0.4720 


0.8816 


0.5354 


1.8676 


1.0792 




50 




10 


0.6836 


0.6316 


0.7753 


0.8146 


1.2276 


0.8872 




50 




20 


0.4945 


0.4746 


0.8802 


0.5392 


1.8546 


1.0763 




40 




20 


0.6865 


0.6338 


0.7735 


0.8195 


1.2203 


0.S843 




40 




30 


0.4974 0.4772 


0.8788 


0.5430 


1.8418 


1.0734 




30 




30 


0.6894 


0.6361 


0.7716 


0.8243 


1.2131 


0.S814 




30 




40 


0.5003 


0.4797 


0.8774 


0.5467 


1.8291 


1.0705 




20 




40 


0.6923 


0.6383 


0.7698 


0.8292 


1.2059 


0.8785 




20 




50 


0.5032 


0.4823 


0.8760 


0.5505 


1.8165 


1.0676 




10 




50 


0.6952 


0.6406 0.7679 


0.8342 


1.1988 


0.8756 




10 


29 





0.5061 


0.4848 


0.8746 


0.5S43 


1 .8040 


1 .0647 


61 





40 





0.6981 


0.6428 0.7660 


0.8391 


1.1918 


0.8727 


5° 







10 


0.5091 


0.4S74 


0.8732 


0.5581 


1.7917 


1.0617 




SO 




10 


0.7010 


0.6450 


0.7642 


0.8441 


1.1847 


0.869S 




50 




20 


0.5120 


0.4899 


0.8718 


0.5619 


1.7796 


1 .0588 




40 




20 


0.7039 


0.6472 


0.7623 


0. 849 1 


I.177S 


0.S66S 




40 




30 


0.5149 


0.4924 


0.8704 


0.5658 


1.7675 


1.0559 




30 




30 


0.7069 


0.6494 


0.7604 


0.8541 


.1.1708 


0.8639 




30 




40 


0.5178 


0.4950 


0.86S9 


0.6696 


1.7556 


1.0530 




20 




40 


0.7098 


0.6517 


0.7585 


0.8591 


1. 1640 


0.8610 




20 




50 


0.5207 


0.4975 


0.8675 


0.5735 


'■7437 


1.0501 




10 




50 


0.7127 


0.6539 


0.7566 


0.8642 


1.1571 


0.85S1 




10 


30 





0.5236 


0.5000 


0.8660 


0.5774 


1.7321 


1.0472 


60 





41 





0.7156 


0.6561 


0.7547 


0.8693 


1.1504 


0.8552 


49 







10 


0.5265 


0.5025 


0.8646 


0.5S12 


1.7205 


1.0443 




50 




10 


0.7185 0.6583 


0.7528 


0.8744 


1.1436 


0.8523 




50 




20 


0.5294 


0.5050 


0.8631 


0.5S51 


1.7090 


1.0414 




40 




20 


0.7214 


0.6604 


0.7509 


0.8796 


1.1369 


0.8494 




40 




30 


0.5323 


0.5075 


0.8616 1 0.5890 


1.6977 


1.0385 




30 




30 


0.7243 


0.6626 0.7490 


0.8847 


1.1303 


0.8465 




30 




40 


0.5352 


0.5100 


0.8601 


0.5930 


1.6S64 


1.0356 




20 




40 


0.7272 


0.6648 


0.7470 ! 0.8899 


1.1237 


0.8436 




20 




50 


0.5381 


0.5125 


0.8587 


0.5969 


1.6753 


1.0326 




10 




50 


0.7301 


0.6670 


0.7451 


0.8952 


1.II71 


0. 8407 




10 


Angle. 


arc. 


cosine. 


sine. 


cot. 


tan. 


arc. 


Angle. 


Angle. 


arc. 


cosine. 


sine. 


cot. 


tan. 


arc. 


Angle. 



THE CONSTRUCTOR. 



299 



ANGLE. 


arc. 


sine. 


cosine. 


tan. 


cot. 


arc. 


ANGLE. 


ANGLE. 


arc. 


sine. 


cosine. 


tan. 


cot. 


arc. 


ANGLE. 


deg. 


min. 






deg. 


rain. 


deg. 


min. 




1 




deg. 


min. 


42 
43 



10 
20 

30 
40 

SO 


10 
20 

30 
40 

50 


0.7330 
0.7359 
0.7389 
0.7418 

0.7447 
0.7476 

0.7505 
0.7534 
0.7563 
0.7592 
0.7621 
0.7650 


0.6691 
0.6713 
0.6734 
0.6756 
0.6777 
0.6799 

0.6820 
0.6841 
0.6S62 
0.6884 
0.6905 
0.6926 


0.7431 
0.7412 

0.7392 
0.7373 
0.7353 
0.7333 

o.73'4 
0.7294 

0.7274 
0.7254 

0.7234 
0.7214 


0. 9004 
0.9057 
0.91 10 
0.9163 
0.9217 
0.9271 

0.932s 
0.9380 

0.9435 
0.9490 

0.9545 
0.9601 


1.1106 
1.1041 
1.0977 
1. 0913 
1.0S50 
1.0786 

1.0724 
r.o66i 
1.0599 
1.0538 

1.0477 
1. 0416 


0.8378 
0.8348 
0.8319 
0.8290 
0.8261 
0.8232 

0.8203 
0.8174 
0.8145 
0.8116 
0.80S7 
0.8058 


48 
47 



50 
40 
30 
20 
10 


so 
40 
30 
20 
10 


44 
45 




lO 

20 
30 
40 
50 




0.7679 

0.7709 

0.7738 
0.7767 

0.779S 
0.7824 

0.7854 


0.6947 
0.6967 
0.6988 
0.7009 
0.7030 
0.7050 

0.7071 


0.7193 
0.7173 
0.7153 
0.7133 
0.7112 
0.7092 

0.7071 


0.9657 
0.9713 
0.9770 
0.9827 
0.9SS4 
0.9942 

1. 0000 


1.035 s 
1.0295 

1.0235 

1. 0176 
I.0II7 

1.0058 

1. 0000 


0.S029 

0.7999 
0.7970 
0.7941 
0.7912 
0.7883 

0.7854 


46 



50 
40 
30 
20 
10 






Angle. 


arc. 


cosine. 


sine. 


cot. 


tan. 


arc. 


Angle. 




ang.=o° \' 


o°5' 


135° 


iSo° 
3.1416 


225° 
3.9270 


270° 
4.7124 


315° 
5.4978 


360° 


Angle. 


arc. 


cosine. 


sine. 


cot. 


tan. 


arc. 


Angle. 


arc. 


= 0. 


0003 


0.0015 


2.3562 


6.2 


S32 



TRIGONOMETRICAL FORMULA. 

sin (a d= /3) = sin a cos j} ± cos a sin J3 

cos (a ± /3) = cos a cos i3 =F sin a sin /? 

sin 2 a = 2 sin a cos a 

sin 30^3 sin a — 4 siti a-' = sin a (4 cos a? — i) 

cos 2 a ^ cos a'^ — sin a' = 2 cos a? — i ^ i — 2 sin a? 

cos 3 u ^ 4 cos a-'' — 3 cos a = cos a (i — 4 sin a') 

a -I- /3 a — .3 

si?i a -\- sin p ^ 2 sifi cos 



sin a — sin p ^ 



a4- /3 . 

2 cos sin 

2 



cos a 4- cos p ^ 2 cos cos 



2 

a — 13 
2 

a — j3 



II. 

12. 
13. 

14. 

15. 

16. 

17. 

IS. 

19. 



. a 4- 13 . 13 — a 

COS a — cos 3 = 2 sin sin 

22 

sin a' = ^^ (i — cos 2 a) 
cos a' = yi (i + cos 2 a) 
sin q' ^ X (3 .f"^ " — -S"' 3 ") 
cos a^ = % (3 '^"^ "■ + ^os 3 a) 

iang a ± tang j3 
I q= iang a tang /3 

colang a cotang /? =F l 



tang (a ± 
cotang (a ± /3) - 
tang 2 a- 



^ cotang a + cotang ,■? 
2 tang a 



cotang 2 n = 
ian^ a 



— tang a' 
cotans: a'' - 



V 



.^s/S 



2 cotang a 



• cos 2 a 



cotang 

tang a zb iang /3 = 



cos 2 a 
cos 2 a 



1 + 2 cos a 
sin 2 a 



I — cos 2 a I — cos 2 a 

sin (a zt /3) 



2 tang Yz "■ 

1 — iang ]A, a^ 

cotang )4 a- — I 

2 cotang yi a 



22. cotang a ± cotang ft = 



cos a cos /3 
sin (/? ± a) 



23. 



sin a sin J3 

sin a 4- sin j3 iang }4 (a + (3) 

sin a — sin j3 tang j^ (a — /?) 



300 



THE CONSTRUCTOR. 





TABL'E OF NUMBERS.-I. 


» 


i 
I 

n 


■,i>- 


«^ 


v/,7 


1 


^,T 


1 


^,~ 


I 
^;7 


0.30 

0-375 
0.60 
0.625 
0.70 


3-333 
2.667 
1.667 
1.600 
1.429 


0.090 
0.141 
0.360 
0.391 
0.490 


0.027 
0.053 
0.216 
0.244 
0.343 


0.548 

0.612 

0.775 
0.791 

0.837 


1.826 

1.633 

1.291 
1.2(35 
1.195 


0.669 
0.721 
0.843 
0.855 
0.888 


1.495 
1.387 
1.186 
1.170 
1,126 


0.740 

0.783 
0.880 
0.889 
0.915 


1.351 
1.278 
1.136 
1.125 
1.093 


0-75 
0.875 
0.90 
1. 10 
1.2 


1-333 
1. 143 
I. Ill 
0.909 
0.833 


0.563 
0.766 
0.810 
1. 210 
1.440 


0.422 
0.670 
0.729 

1.331 

1.728 


0.866 
0.935 
0.949 
1.049 
1.095 


1.155 
1.069 
1.054 
0.953 
0.9 '3 


0.909 
0.956 
0.965 
1.032 
1.063 


1. 100 
1,046 
1.036 
0.969 
0.941 


0-931 
0.974 
0.987 
1.024 
1.047 


1.075 
1.024 
1.013 
0.976 
0.955 


1.25 
1.50 

1-75 
2.0 

2.25 


0.800 ' 

0.667 

0.5-1 

0.500 

0.444 


1.563 

2.250 

3.063 

4.0 

5.063 


1.953 
3.375 
S-359 
8.0 

11.391 


1. 118 
1.225 

1.323 
I.4I4 
1.500 


0.894 
0.816 
0.756 

0.707 
0.667 


1.077 

1.145 
1.205 
1.260 
1.310 


0.928 

0.874 
0.830 
0.794 
0.763 


1.057 
1.107 
1.150 
1.189 
1.225 


0.946 
0.904 
0.869 
0.841 
0.816 


2.50 

2.75 

3.0 

3-25 

3.50 


0.400 
0.364 
0.333 
0.308 
0.286 


6.250 

7.563 

9.0 

10.563 
12.250 


15.625 
20.797 
27.0 
34.328 

42.875 


1.581 
1.658 

1.732 
1.803 
1.871 


0.632 
0.603 
0.577 
0.555 
0.53s 


1.357 
1.401 
1.442 
1.481 
1.518 


0.737 
0.714 
0.693 
0.675 
0.659 


1.257 
1.2S8 
1.318 
1.342 
1.368 


0.795 
0.777 

0.759 
0-745 
0-731 


3-75 

4.0 

4.5 

5.0 

5-5 


0.267 
0.250 
0.222 
0.200 
0.182 


14.063 

16.0 

20.250 

25.0 

30.250 


52.734 

64.0 

91.125 
125.0 
166.375 


1.936 

2.0 

2. 121 
2.236 
2.345 


0.516 
0.500 
0.471 

0.447 
0.426 


1-554 
1.587 
1. 651 
1.710 
1.765 


0,644 
0.630 
0.606 
0.585 
0.567 


1.392 
1.414 
1.457 
1.495 
1.531 


0.719 
0.707 
0.6S7 
0.669 
0.653 


6.0 

6.5 
7.0 

7-5 
8.0 


0.167 
0.154 
0.143 
0-133 
0.125 


36.0 

42.25 
49.0 
56.250 
64.0 


216.0 

274.625 

243.0 

421.875 

512.0 


2.449 
2.550 
2.646 

2.739 
2.828 


0.408 
0.392 
0.378 
0.365 
0.354 


1.817 
1.866 
1.913 
1.957 
2.0 


0.550 
0.536 
0.523 
0.510 
0.500 


1.565 
1.597 
1.627 

1.655 
1.682 


0.639 
0.626 
0.615 
0.604 
0595 


8.5 
9.0 
9-5 

IC 

II 


0.1 iS 
0.1 1 1 
0.105 
0.1 00 
0.091 


72.250 

81.0 

90.250 
1 00.0 
121.0 


614.125 

729.0 

857-375 

lOOO.O 

I33I-0 


2.915 
3.000 
3.082 
3.162 
3.317 


0.343 
0.333 
0.324 
0.316 
302 


2.041 
2.080 
2.118 
2.154 
2.224 


0.490 
0.481 
0.472 
0.464 
0.450 


1.707 
1.732 
1.756 
1.778 
1.821 


0.586 
0.577 
0.570 
0,562 
0.549 


12 

13 
14 
15 
16 


0.083 
0.077 
0.071 
0.067 
0.063 


144 
169 
196 
225 
256 


1728 
2197 
2744 

3375 
4096 


3-464 
3.606 

3.742 

3.873 
4.000 


0.289 
0.277 
0.267 
0.258 
0.250 


2.289 

2.351 
2.410 
2.466 
2.520 


0.431 
0.425 
0.415 
0.405 
0.397 


1.861 
1.899 
1.934 
1.968 
2.000 


0.537 
0.527 

0.517 
0.508 
0,500 


17 
18 

19 
20 
50 


O.OS9 
0.056 

0.053 
0.050 
0.020 


289 
324 
361 
400 
2500 


4913 
5832 
6S59 
8000 
125000 


4.123 
4.243 
4.359 
4.472 
7.071 


0.243 
0.236 
0.229 
0.224 
0.141 


2.571 
2.621 
2.668 

2.714 
3.684 


0.389 
0.381 

0.375 
0.368 
0.271 


2.031 
2.060 
2.088 
2.115 
2.659 


0.492 
0,485 
0.479 
0-473 
0-376 


100 
1000 
n- = 3.142 

2 ff ^ 6.2S3 


0.0 10 
0.00 1 
0.318 
0.159 


1 0000 
I 000000 

9.870 
39-478 


I 000000 

I 000000000 

31,006 

248.050 


lO.O 

31.623 
1.772 
2.507 


0.10 
0.032 
0.564 
0-399 


4.642 
1 0.0 
1.465 
1845 


0.215 
O.I 00 
0.683 
0.542 


3.162 
5.623 
1-331 
1.583 


0.316 
0.178 
0.751 
0.632 


-=1.571 


0.637 


■ 2.467 


3-878 


1.253 


0.798 


1.162 


0.860 


1.120 


0.893 


■K 

~ = 1.047 


0.955 


1.097 


1. 148 


1.023 


0.977 


1.016 


0.985 


1.012 


0.989 


4 

- TT = 4.189 


0.239 


17.546 


73.496 


2.047 


0.489 


1. 612 


0.622 


1.431 


0.699 


-=0.785 
4 


1.274 


0.617 


0.484 


0.886 


1.128 


0.923 


1.084 


0.941 


1.062 


^ = 0.524 


1. 910 


0.274 


0.144 


0.724 


1.382 


0.806 


1. 241 


0.851 


1. 176 


rr"- = 9.S70 


O.IOI 


97.409 


961.390 


3.142 


0.318 


2.145 


0.466 


1.772 


0.564 


7r' = 31.006 


0.032 


961.390 


29809.910 


5.568 


1.796 


3-142 


0.318 


2.360 


0.424 


= 0.098 

32 


10.1S6 


0.0095 


O.OOI 


0.313 


3.192 


0.461 


2.1 68 


0.560 


1.782 


3'^ 

76= °-5«9 


1.698 


0.347 


0.204 


0.768 


1.303 


0.S38 


1.194 


0.876 


1. 142 


g = 32.2 
2^ = 64.4 


0.031 
0.015 


1036.84 

4147-36 


33386.24 

267090 


5.674 

8.025 


0.176 
0.125 


3,181 
4,007 


0.314 
0.249 


2.381 
2.833 


0,419 
0-337 



THE CONSTRUCTOR. 



301 











TABLE OF NUMBERS.- 


-II. 








n 


VTi 


\ 


n 


v/17 


^T 


n 


x/17 


i 


n 


v/^ 


^17 


0.0 1 


0.1 0000 


0.21544 


0.26 


0.50990 


0.63825 


0.51 


0.71414 


0.79896 


0.76 


0.87 1 78 


0.91258 


0.02 


0.14132 


0.27144 


0.27 


0.51962 


0.64633 


0.52 


0.72111 


0.80415 


0.77 


0.S7750 


0.91657 


0.03 


0.17321 


0.31072 


0.28 


0.52915 


0.65421 


0-53 " 


0.72801 


0.80927 


0.7S 


0.88318 


0.92052 


0.04 


0.20000 


0.34200 


0.29 


0.53852 


0.66191 


0.54 


0.73485 


0.81433 


0.79 


0.8S882 


0.92443 


0.05 


0.22361 


0.36840 


0.30 


0.54772 


0.66943 


0-55 


0.74162 


0.81932 


0.80 


0.89443 


0.92832 


0.06 


0.24495 


0.39149 


0.31 


0.55678 


0.67679 


0.56 


0.74833 


0.82426 


o.Si 


0.90000 


0.93217 


0.07 


0.26458 


0.41213 


0.32 


0.56569 


0.68399 


0.57 


0.75498 


0.82913 


0.82 


0.90554 


0-93599 


o.c8 


0.2S2S4 


0.43089 


0-33 


0.57446 


0.69104 


0.58 


0.76158 


0.83396 


0.83 


0.91104 


0.93978 


0.09 


0.30000 


0.44814 


0-34 


0.58310 


0.69795 


0.59 


0.768 1 1 


0.83872 


0.84 


0.91652 


0.943S4 


O.IO 


0.31623 


0.46416 


0.35 


0.59161 


0.70473 


0.60 


0.77460 


0.84343 


0.85 


0.92195 


0.94727 


0.1 1 


0.33166 


0.47914 


0.36 


0.60000 


0.71138 


o.5i 


0.78102 


0.84809 


0.86 


0.92736 


0.95097 


0.12 


0.34641 


0.49324 


0.37 


0.60828 


0.71791 


0.62 


0.78740 


0.85270 


0.87 


0.93274 


0.95464. 


0-13 


0.36056 


0,5065s 


0.38 


0.61644 


0.72432 


0.63 


0.79373 


0.85726 


0.88 


0.93808 


0.95828 


0.14 


0.37417 


0.51925 


0-39 


0.62450 


0.73061 


0.64 


0.80000 


0.86177 


0.89 


0.94340 


0.96190 


0.15 


0.3S730 


o-53'33 


0.40 


0.63246 


0.73681 


0.65 


0.80623 


0,86624 


0.90 


0.94868 


0.96549 


0.16 


0.40000 


0.542S8 


0.41 


0.64031 


0.74290 


0.66 


0.81240 


0.87066 


0.91 


0.95394 


0.96905 


O.I7 


0.41 231 


0-55397 


0.42 


0.64807 


0.74889 


0.67 


0.81854 


0.S7503 


0.92 


0.95917 


0.97259 


0.18 


0.42426 


0.56462 


0.43 


0.65574 


0-75478 


0.68 


0.S2462 


0.87937 


0.93 


0.96437 


0.97610 


0.19 


0.43589 


0.57489 


0.44 


0.66332 


0.76059 


0.69 


0.83066 


0.88366 


0.94 


0.97954 


0.97959 


0.20 


0.44721 


0.58480 


0-45 


0.67082 


0.76631 


0.70 


0.83666 


0.8S790 


0.95 


0.97468 


0.9S3O5 


0.21 


0.45826 


0.59439 


0.46 


0.67823 


0.77194 


0.71 


0.84261 


0.S9211 


0.96 


0.97980 


0.9864S 


0.22 


0.46904 


0.60368 


0.47 


0.6S557 


0.77750 


0.72 


0.84853 


0.89628 


0.97 


0.98489 


0.98990 


0.23 


0.47958 


0.61269 


0.48 


0.69282 


0.78297 


0.73 


0.85440 


0.90041 


0.98 


0.98995 


0.99329 


0.24 


0.48990 


0.62145 


0.49 


0.70000 


0.78S37 


0.74 


0.S6023 


0.90450 


0.99 


0.99499 


0.99666 


0.25 


0.50000 


0.62996 


0.50 


0.7071 1 


0.79370 


0.7S 


0.86603 


0.90856 


1. 00 


1. 00000 


1. 00000 






sin 30 


° = cosi> 


3° = K; 


CO. 


f 30° = SI 


« 60° = j4 


\/^= 0.8660. 










sin 75 


° = cos\ 


5° = 0.9659 


; ta 


I So° = c 


jl 60° = 'Ji 


^T= 0.5774 ; 










cosT^ 


° = sin I 


5° = 0,258s 


; CO 


'an 30° = 


(an 60° = 


^T= 1.73 


21. 










log TV 


= 0.49714 


99- 


log 


'.§•=1.5 


37856. 













^LF'H^BETIC^L UsTDEX. 



ACCUMULATORS 264 

Hoppe's 264 

" Hydraulic 218 

Tweddells' . . . . 265 

Action of GearTeeth 129 

Adamsou's Stiffening Ring 269 

Addition and Subtraction of Forces 26 

Addition, Graphical 26 

Adjustable Escapements 170 

" Gears for Rotative Mot- 
ors 237 

" Hangers 74 

" Power Escapements . . . 236 

" Pump Gears 236 

Admiralty Chain 1S2 

Adyman's Coupling 215 

Agadio's Cable Locomotive 176 

Air Compressors, Riedler's 279 

Air Pump, Bunsen's 222 

" Voa Gerike's 225 

" Watt's 225 

Air, Reservoir for Compressed 272 

Allan's Link Motion 235 

Almgren's Researches on Steam 

Boilers 271 

Althaus' Furnace Hoist 173 

Althaiis' Pump 223 

American Standard Car Bearing. ... 75 

Amos & Smyth's Pump 224 

Anchor Escapement, Free 168 

Bolts 56 

Ratchet I55 

Anemometers 239 

Angle and T Iron Columns 83 

Angle of Torsion, Determination of 93 

Angle of Rotation in Torsion 11 

Angstrom's Valve Gear 233 

Anti-Friction Wheels 123 

Anti-parallel 22 

Anti-projection 23 

Application of Tension Organs 172 

Archimedes, Tympanon of 221 

Archimedian Screw 221 

Area o£ Polygons 23 

' ' Quadrilaterals 24 

Triangles 23 

Arithmography 22 

Arm Sections, Table for Transform- 
ing 103 

Arms of Gear Wheels 149 

Armstrong Hydraulic Crane 22S 

Artificial Draft 272 

Atmospheric Railway 227 

Attachment of Journals 67 

Audemar's Pump 224 

Automatic Coupling loi 

" Friction Brake 170 

" Steam Stop 281 

Axis, Neutral 3 

Axle, American Railway Standard.. 89 

Prussian Railway Standard. . . 89 

" Simole Crank 106 

Axles .'. 85 

" for Water Wheels 91 

" Graphical Calculation of 86 

" Loaded at Two Points 87 

Axles 107 

" Non-Symmetrical 86 

" Proof Diagrams of 87 

" Proportions of 86 

" Railway 88 

" Symmetrical 85 

" with Circular Section 85 

" with Cruciform Section 90 

" with Inclined Loads 90 



Axles with Three or more Bearings . 89 
' ' Wooden 92 

BAG PUMP 217 

Baker's Blower 221 

Balanced Valves 279 

Balance Wheel 167 

Balancing of Pulleys 194 

Balanced Slide Valves 285 

Balanced Valve, Cramer's 280 

Ball Bearings 127 

Ball Joints for Pipes 249 

Band Saws 177 

Base Figures for Hyperboloidal 

Wheels 136 

Bastard Gears 135 

Beale's Gas Exhauster 226 

Beams 3 

' ' Double Trussed 35 

Sections, Table of 5-7 

Force Plans for Framed 38 

Multiple Trussed 35 

Scale Ill 

" Simple Trussed 35 

Triple Trussed 35 

Walking no 

" with Common Load 11 

Bearings, Ball 127 

Design and Proportion of 68 

" Independent Step 75 

" Lateral 68 

Metaliue 179 

Multiple Collar 77 

Multiple Supports for Fo 

Pedestal 71 

' ' Roller 126 

Roller for Bridges 126 

Simple Supports for 79 

Standard American Car. . . 75 
" Standard Prussian Car. ... 75 

" Special Forms of 74 

Step 75 

Bearing Supports, General Principles 82 

Bearings, Thrust '. . . . 65, 68, 75 

" Thrust with Wooden Sur- 
face 76 

' ' Supports for 79 

Wall 68, 71 

Wall Step 75 

with Three-part Boxes ... . 70 

" Yoke 72 

Becker's Clutch loi 

Behren's Chamber Gear Train 220 

Belidor's Water Pressure Engine . 229 

Bell Crank no 

Bellegarde, Rope Transmission at , . 205 

Belleville Elastic Washers 212 

Bellows Pump 217 

Bell Pipe Connections 248 

Bell Valve '. 276 

Belt Connections 191 

" Fastening, Bolter's 193 

" Fastening, Moxon's 193 

Belting 186 

' ' Cement for 193 

" Efficiency of 194 

Specific Capacity of 190 

' ' Stress on 191 

" Various Examples of 1S7 

" Table of Examples 192 

Belt Lacing 191 

Belts, Capacity of 190 

" Creep of 194 

" Cross Section of igo 

" Path of 186 



Belts, Polishing 1 177 

Quarter Twist 186 

Belt Shifters ■. 188 

Belt Shifter, Zimmerman's iSg 

Belts, Stiffness of 194 

Belt Transmission, Examples of.... 191 

Belts, Transporting 221 

Bending, Bodies of Uniform Resist- 
ance to 8 

" Load 3 

Moment 3 

' ' Resistance to 2 

Bergner's Drawing Board 172 

Berlin, Sewerage System of 219 

Bevel Friction Wheels 124 

" Friction Wheels, Minotto's... 125 

Gears 135 

" " Construction Circles for. 135 

" " Spiral 141 

'■ " Stepped 141 

Beylich's Universal Gears 136 

Biquadratic Parabola 10 

Blake's Steam Pump 230 

Bleichert's Cable Tramway System . . 175 

Blower, Baker's 221 

Blower, Root's 221 

Blowers, Fan 222 

Bloxam's Gravity Escapement 168 

Boat, Sail 223 

Boat, Chain Propulsion of 183 

Bodies of Uniform Strength 2 

Bogardus Mill 126 

Boiler ConstructioB, Economy in 

Combustion. 270 
" " Economy of 

Material in 270 
'■ " Improvements 

in Heating 
Surface . . . 270 

Boiler Details 266 

" Feeder, Brindley's 228 

Feeders 228 

Flues 269 

' ' Flues, Corrugated 269 

Riveting 42 

Boilers, Almgren's Researches on.. 271 
" Circumferential Seams of.. 268 

" Classified 265-266 

Flat Surfaces of 268 

for Swedish State Railway. 272 

Longitudinal Seams of 267 

Thickness of 266 

" Spherical Details 26S 

' ' Steam 265 

Boiler Tubes 270 

Boiling Water, used for Shrinking. . 47 

Bolt Connections, Unloaded 60 

" Dead 166 

" Gerber's 57 

" Heads 54 

" Latch 166 

Bolts, Anchor 56 

" and Nuts, Metric 55 

" and Screws 50 

" Maudslay's Method of Secur- 
ing 58 

Parsons' 57 

Penn's Method of Securing.. 57 

" Special Forms of 55 

Borda Turbine 220 

Bored Guides 122 

Botter's Belt Fastening 193 

Boxes, Various Forms of Journal ... 6g 

Brace, Weston's Ratchet 154 

Bracket Support for Bearings 7q 



304 



ALPHABETICAL INDEX. 



Brackets, Wall 72 

Brake, Automatic Friction 170 

Brake, Napier's Differential 214 

Brakes, Chain 215 

Clamping 214 

" Internal Strap 215 

" Sliding 215 

" Strap 211-214 

Brake, Toggle Friction 162 

Bramah Lock 166 

Brasses for Connecting Rod 112 

Brauer's Intermittent Gearing 165 

Breaking Load i 

Bridge Bolts 59 

Flying 222 

" Roller Bearings 126 

" St. Louis 60 

Briggs' System of Pipe Threads. . . . 250 

Briadley's Boiler Feeder 22S 

Britton's Steering Gear 23S 

Brown's Valve Gear 235 

Brown's Windlass 173 

Buckling, Resistance to 13 

Buckling Strains, Table of 14 

Buffer Coupling? iSi 

Built up Screw Propellers 57 

Bnnsen's Air Pump 222 

Butler's Coupling 96 

CABLE, Arrangement of Pulleysfor 202 

" Drum, Fowler's 173 

" Ferry System, Hartwich's. 175 

Grip Pawl 185 

" Haulage Systems 174 

" Incline at Lucerne 173 

" Locjrnotivi, Agudio's . . . . 176 

" Railways. 173 

" Rhenish Railway 174 

" Road, Kahlenberg 173 

Cables, Table for Tightened 200 

Cable System for Canals, Schmick's. 175 

" System. Riggenbach's 174 

" Tramway, Chicago 175 

" Tra-nw-iy, Overhead 175 

" Tramways, San Francisco. . . . 174 
" Transmission, Ring System 208-211 
" Transmissions, Short Span... 200 
" TransmissioQ with Inclined. . . 200 

Cadiat Tnrbin 3 220 

Cagaiardelle 221 

Call & Co., Valve Gear 162 

Calculating Machine, Thomas'. . .153, 156 

Calculation of Springs 18 

Calculations for Chains 183 

Cambon's Roller Bearing 127 

Cam Valve gears 236 

Canal Cable System, Schmick's 175 

" Lift at Les Fontinettes 227 

■' " Green's 227 

" " La Louviere 227 

" " Mersey 227 

Locks " 227 

Cannon, Thickness of 15 

Capacity of Belts 190 

Capstan, David's 173 

Car Bearing, American Standard... 75 
Car Bearings, Prussian Standard ... 75 

Cardan's Coupling 97 

Cardioide 92 

Casting 240 

Cast Iron Cranks 105 

Central Curve of Valve Gear 235 

Centre of Gravity Graphically De- 
termined 33 

Centrifugal Force 177 

" Force of Wire Rope.... 197 

" Pumps 222 

Chain, Admiralty 182 

" Brakes 215 

" Couplings 184 

" Drums 185 

Flat Link 1S3 

" Gemorsch 182 

" Madgeburg-Bodenbacher. . . . 183 

" Open Link 182 

Pawls for 185 

Pitch 183 



Chain Propeller, Heuberger's 176 

Propelling Gear 187 

" Proportions of 183 

" Propulsion of Boats 183 

Chains, Calculations for 183 

Chain Sheaves 1S5, 211 

Chain, Specific Capacity of 211 

Chains, Running 1S2 

Stationary 182 

' ' Tests for 183 

Chain Strippers 1S5 

Swivels 184 

" Transmission 211 

" Transmission, Efficiency of... 213 

in Mines 213 

" " Decido Mines. 212 

" Weight of 183 

Chamber Gear Train, Behren's 220 

" " " Eve's 220 

" " " Repsold's. . . . 220 

" Wheel Trains 2ig 

Channeled Connecting Rods 117 

Checking Ratchets 150, 163 

Check Valves 274 

Cheese Coupling 99, 151 

Chemical Ratchet Trains 171 

Cheret's Press, Friction Gear of. . . . 125 

Chicago Cable Tramway 175 

Chronometer Escapement 167 

Chubb Lock 166 

Circular Plate, Deflection of 15 

Circumference Scale 12S 

Clamp Coupling 95 

Clamping Brakes 214 

Clamp Pulley, Fowler's 203 

Clamp Ratchet 160 

Clark's Canal Lift 227 

Clerk, Method of Shrinking Rings. 45 
Clocks, Striking Mechanism for, . . . 169 

Close Link Chain 182 

Clutch, Becker's loi 

" Cone 99 

" Couplings 95,98 

" Coupling, Fossey's 100 

" Dohmen-Leblanc's 101 

" Forks 99 

Clutches, Friction 99 

Clutch, Garaud's loi 

" Jackson's loi 

" Koechlin's Friction 100 

" Napier's loi 

" Reuleaux's Friction 100 

" Schurmann's loi 

Toothed 98 

" Weston's Friction loi 

Coating Operations 241 

Cock, Four Way 225 

Cocks 281 

Coefficients of Resistance i 

Coefficients of Safety 1 

Cold Forcing 17 

Forcing, Dimensions for 47 

-" Hooping 45 

Collar Thrust Bearings 66 

Columns, Calculations for Iron 82 

Fluted 83 

" Forms for Iron 84 

" Grouped 84 

' ' Hollow 83 

of Angle and T Iron 83 

" of Uniform Resistance.... 13 

" Strength of Cast Iron 83 

Stresses in 82 

Combined Levers no 

Compound Escapements 168 

" Link as Thrust Bearing. 67 

" Strains, Table of 15 

" Stresses 13 

Compressed Air for Power Distribu- 
tion 219 

Compression, Resistance to 2 

Condenser, Watt's 230 

Conditions of Equilibrium 29 

Conductors for Pressure Organs 242 

Conduits for Pressure Organs 216 

Cone Clutch Coupling 99 



Cone Coupling, Reuleaux's 96 

" Pulleys 189 

" " Diagram for igo 

" " for Crossed Belts iSg 

" " for Open Belts iSg 

Conical Gear Wheels 135 

Connecting Rod Brasses 112 

" End, Cast Iron.... 113 

'■ " Krauss' 113 

" " Penn's 113 

" " Polonceau's. . 114 

" Porter- Allen 117 

Rods J12 

" Channeled 117 

" Forms of ij8 

" Locomotive 116 

" Rectangular Section 117 

Rod, Solid End for 113 

Rod, Solid End for Lo- 

motive 113 

Rods, Ribbed 117 

Rods, Round ij6 

Red, Strap End for 112 

Rod, Whip Action of . . . . 116 

Connections for Belting igi 

" Cast Iron Pipes.... 248 

" Crank Axles 115 

" Lead Pipe 251 

•' Neck Journals 114 

" Wrought Iron Pipes 249 
Construction Circles for Bevel Gears 135 
Construction of Machine Elem'nts. 39-289 

" Pulley Stations 204 

" Rope Curve 202 

" Rope Pulleys 202 

Screw Thread 50 

Continuous Ratchets 150 

Ratchets with Locking 

Teeth 165 

Running Ratchets 164 

Copper Pipes 246 

Cord Friction 177 

Cord Polygon 26 

Corliss Valve 236 

Corliss Valve Gear 162 

Cornish Valve 280 

Cornish Valve Gear 153, 163 

Corrugated Boiler Flues 269 

Corrugated Fire Box 269 

Cotton Rope 179 

Cotton Rope Transmission 196 

Counterbalance, Oeking's .... 217 

Countershaft Hanger, Sellers' 74 

Counting Gear for Gas Meter 165 

Couples, Force 29 

Coupling, Adyman's Z15 

" Butler's 96 

" Cardan's 97 

" Cheese 99, 151 

" Clamp 95 

" Cresson's 96' 

" Drag Link 97 

' ' Hooke's 97 

" Muff 95 

" Oldham's 96 

Plate 95 

" Pouyer's loi 

" Prentiss 216 

" Ramsbottom's Friction... 99 

" Reuleaux's Cone 96 

Couplings 95 

" Automatic loi 

" Buffer 181 

" Schurman's Friction 215 

Clutch.... 98 

Coupling, Sellers' 96 

Couplings, Flexible 96 

" for Chain 184 

' ' for Propeller Shafts 95 

" Sharp's 96 

" Link gS 

Coupling, Uhlhorn's loi 

Cramer's Balanced Valve 2S0 

Crane Hook, Proportions for 184 

" Pillars 89 

' ' Ramsbottom's 176 



ALPHABETICAL IXDEX. 



Cranes, Cotton Rope Driven 196 

Graphical Calculation 27 

" Hydraulic 228 

Squaring Device for 172 

" Varieties of 173 

Crane, Tangye's -^ \lb 

Crane, Towne's 176 

Crank Axle, Graphostatic Calcula- 
tion of 106 

" Axles, Connections for 115 

" Axle, Simple 106 

" Graphostatic Calculation for 

Return 105 

" . Graphostatic Calculation for 

Single 104 

" Pin, Tangential Pressure on. 233 

Pins 61 

" Pins, Connections for 112 

" Return 105 

Cranks, Cast Iron 105 

Cranks, Classified 104 

Crank Shaft, Graphostatic Calcula- 
tion of 107 

Cranks, Hand 109 

Crank, Sliding 226 

Cranks, Single Wrought Iron 104 

Creep of Ropes 196 

Creep of Belts 194 

Cresson's Coupling 96 

Crossed Belts, Cone Pulleys for 189 

Cross Heads 118 

" for Guides 119 

" " for Link Connections... 119 

" " for Locomotives 121 

" " for Marine Engines.... 120 

" Head, Slipper 121 

" Head, Superficial Pressure on. 120 

'• Keyed Connections 48 

" Section of Belts 190 

" " Hemp Rope 195 

" " Wire Rope 196 

Crown of Pulleys 186 

Ratchet 154 

Wheel Escapement 169 

Cup Packing 253 

Current Motor 223 

Curve, Elastic 3 

Curves, Velocity 233 

Cycloidal Curves 130 

" Curves, Generation of 130 

" Sinoide 13 

Cycloid, Spherical I35 

Cylinder Escapement 169 

Ratchet 156 

" Ratchet Gearing 165 

Cylinders 216 

Cylinders for Hydraulic Presses 243 

Cylindrical Spiral Gears 138 

Cylindrical Vessels 15 

DANAIDE 220 

Darcy, Formula for Friction of 

Water 246 

David's Capstan 173 

Davis & Co., Steering Gear 238 

Dead Bolt 166 

Dead Ratchet Tooth 152 

Deane's Steam Pump 230 

Decido Mines, Chain Transmission of 212 

Decomposition into Parallel Forces. 31 

Deflection 3 

" in Bodies of Uniform Re- 
sistance 8 

" of Circular Plate 15 

" of Shafting 94 

" of Shafting, Torsional. ... 92 

" of Wire Ropes 198 

Delisle's Screw Thread Systems. ... 53 

Dennison's Escapement 168 

Design and Proportion of Bearings. 68 

Diagram for Cone Pulleys 190 

Diametral Pitch 12S 

Diaphragm Pump 217 

Differential Brake, Napier's 214 

" Hydraulic Lever 218 

Pulley Block, Weston's. 174 



Differential Windlass 

Dimensions of Gear Wheels 

Disk Friction Wheels 

" Valves, Flat 

" Wheels with Pin Teeth 

Distribution of Weight 

Division by Lines 

Division of Gear Wheels, Circumfer- 
ential . . . 

Dobo's Ratchet . 

Dohmen-Leblanc's Clutch 

Donnadieu's Pump 

Door Locks 

Double Acting Pumps 

" Arm Pulleys 

" Beat Valve 

" Friction Ratchets. . . 

" Spiral Gears 

Douglas & Coulson's Steering Gear. 

Downton's Pump 

Draft, Artificial 

Draft Keys. 

Drag Link Coupling 

Drawing Board, Bergner's 

Driving by Tension Organs 

Drop Hammer, Friction 

Drop Hammer, Merrill's 

Drums for Chain 

Dry Gas Meter 

Ducommun & Dubied's Planing Ma- 
chine 

Dunning & Boissiere's Steering Gear 

Duplex Escapement 

Pump, Mazelline's 

Pump, Worthington's 



173 
147 
124 

275 

133 

3 

23 

128 
160 

lOI 

223 
166 
224 

193 
280 

160 
141 
238 
224 
272 
48 

97 

172 

173 
176 
123 

185 
240 

176 
238 
167 
231 
231 



ECCENTRICS 109 

Eccentric Straps 115 

Edge Keys 49 

Efliciency of Belting 194 

" of Chain Transmission... - 213 

" of Rope - Transmission. . . 205 

Equalizing Levers 32, iii 

Equalizer Worthington's 232 

Equation of Elastic Curve 3 

Equatorial Section Modulus... ■ 5 

Equilibrium, Conditions of 29 

" of External Forces. ... 27 

of Forces 22 

" of Internal Forces. ... 28 
of Three Parallel For- 



Elastic Curve 

Elastic Curve, Equation of 

Elasticity and Strength of Flexure . 

Elasticity, Modulus of i, 

Elastic Limit i 

" Limit in Beams 

" Line 92 

Washers, Belleville 212 

Elbe, Chain Propelling Gear on. . . . 185 

Elbow Fittings 251 

Elbow Fittings, Friction in 251 

Electric Signals, Siemens & Halske. 166 

Elements of Graphostatics 22-38 

Elevator, Hydraulic 227 

Elevator Safety Devices 164 

Emery Weighing Machine 173 

Enderlein's Escapement 168 

Engine, Porter-Allen 236 

Engines, Rotative Pressure 233 

Engines, Valve Gear for Rotative. . - 234 

Enlarged Screws 58 

Epicycloidal and Evolute Teeth 

Compared 135 

Erhardt's Flange Joint 47 

Escapement, Bloxam's Gravity 168 

Chronometer 467 

Crown Wheel 169 

Cylinder 169 

Dennison's 168 

Duplex 167 

Enderlein's 168 

Free Anchor 168 

Hipp 168 

Graham's 169 



3o5 

Escapement, Lamb's 168 

Lepaute's 169 

Mudge's 16S 

" Reuleaux's 16S 

Escapements 150, 167 

Adjustable 170 

" Compound 16S 

for Measurements of 

Fluids 239 

for Measurements of 

Volume. . . : 239 

" for Moving Liquids, 

Pressure 228 

" for Pressure Organs. . 226 

Isochronous 167 

" Periodical 169 

" Periodical Pressure. . . 229 

Period of 167 

Power 169 

" Power Adjustable. .. . 237 

Range of 167 

" Simple 167 

Time of Oscillation . ... 167 

Uniform 167 

Escapement, Tiede's 168 

Eve's Chamber Gear Train 220 

Evolute Rack Teeth 132 

Evolute Teeth for Interchangeable 

Gears 131 

Examples of Belting, Table 192 

of Belt Transmission 191 

of Gearing 147 

of Journals 62 

Thrust Bearings 78 

Expansion Gear, Farcot's 236 

'" Gonzenbach's 236 

" Meyer's 236 

" Joints 245 

Valves 236 

External Forces, Equilibrium of . . . . 27 

Extraction of Roots 26 

Eytelwein's Formula for Stiffness of 

Ropes 181, 196 

FABRY'S VENTILATOR 221 

Factor of Safety i 

Fairbairn, Experiments on Boiler 

Flues 269 

Fan Blowers 222 

Farcot's Stuffing Box 254 

Farcot's Valve Gear 236 

Fast and Loose Pulleys 188 

Felbinger's Postal Tube 227 

Ferules for Boiler Tubes 270 

Fink's Link Motion 235 

Fire Box, Corrugated 269 

Fire Box, Kaselowsky's 269 

Fish Torpedo 171 

Flange Connections for Lead Pipe. 252 

Joints 58 

Joint, Erhardt's 47 

Joints for Pipes 248 

Flanges for Riveted Pipes 249, 

Flap Riveted Joints 40 

Flap Valves 274 

Flat Hemp Rope 178 

" Link Chain, Neustadt's 183 

" Link Chain, Table of 183 

' ' Pivot Bearings 66 

" Ropes I Si 

Flexible Couplings 95, 96 

Pipes 252 

Rod Connection 114 

Flexure, Elasticity and Strength of. 2 

Flexure, Strains of 3 

Flow of Metals 240 

Fluid Escapements for Transporta- 
tion 227 

" Running Ratchet Trains 223 

" Valves 287 

Fluted Columns 83 

Flying Bridge 222 

Fly Wheel, Oscillating 233 

Fly Wheels 233 

Force, Centrifugal 177 

Force Couples 29 



3o6 



ALPHABETICAL INDEX. 



Forced Connections, Examples of. . . 46 

Forced Draft 272 

Force Plans for Framed Structures. 34 

Plans for Roof Trusses 36 

" Polygon 26 

Forcing Fil I7 

Forces, Addition and Subtraction of 26 

Equilibrium of 22 

Resultant of Several 26 

Forcing 45 

Forks, Clutch 99 

Forlc Journals 63 

Fork Journal, Stub End for 114 

Forms for Iron Columns 84 

Fossey's Coupling 100 

Foundation Bolts, Keying for 48 

Fourneyron Turbine 220 

Four Way Cock 225 

Fowler's Cable Drum i73 

Fowler's Clamp Pulley 203 

Friction Brake, Automatic 170 

Brake, Toggle 162 

Clutches 99 

Clutch, Koechlin's 100 

Ramsbottom's 99 

Reuleaux's 100 

" Westons loi 

Cord 177 

Coupling, Schurman's 215 

" Drop Hammer ■ 176 

Feed, Sellers' 126 

Gear of Cheret's Press 125 

" Gear, Robertson's 125 

in Elbow Fittings 251 

" in Spur Gearing i34 

in Stufang Box 254 

of Chain Transmission 213 

' ' of Journals 64 

" of Pivot Bearings - ■ 66 

" of Screw Thread 50 

of Spiral Gear Teeth 140 

" of Water in Pipes 246 

Friedmann's Jet Pump 222 

Friction Pawls i59 

Pawl, Saladin's 161 

" Pawls, Release of 161 

Ratchets 158 

Double 160 

Rod 163 

Running 160 

" '■ Stationary 161 

" Rollers, Mechwart's 127 

" Trains, Special .... 126 

Wheels 122 

" Bevel 124 

" " Construction of 123 

" " Disk 124 

" '* for Inclined Axes. . . 124 

" " for Parallel Axes... 123 

" " Material for 123 

" Minotto's 125 

" " Robertson's 125 

" " Sellers' 126 

" " Two Applications of 123 

" " Wedge 125, 160 

Francis' Turbine 220 

Framed Beams, Force Plans for. ... 38 
Framed Structures, Force Plans for 34 
Frankfurt on Main, Water Supply of 218 

Free Anchor Escapement 168 

Free Cross Heads 119 

li'reiburg. Rope Transmission at ... . 205 

French Lock 166 

Front Bearings 6S 

Furnace Hoist, Althaus' 173 

Furnace, Wilson's Water Gas 288 

Future Possibilities of Boiler Con- 
struction 270 

GANNOW MINE 214 

Garand's Clutch loi 

Gases, Reservoirs for 219 

Gas Exhauster, Beale's 226 

" Holders 272 

" Meter, Counting Gear for 165 

" " Dry 240 

Sanderson's 239 



Gas Meter, Wet 239 

Gate Valves 282 

Gearing, Brauer's Intermittent 165 

" Calculation of Pitch and 

Face 144 

" Cylinder Ratchet 165 

" Double Pin 132 

" E.xamples of 147 

" Friction in Spur Tooth. ... 134 
" Fundamental Formula for. 12S 

" Globoid Worm 143 

" Hawkin's Worm; 143 

" Jensen's Worm 143 

Ratchet 1 50 

Shield 133 

" Gearing, Step 141 

Toothed 127 

"' Worm. 139 

Gears, Bastard 135 

Bevel 135 

Bevel Spiral 141 

Beylich's Universal 136 

Cylindrical Spiral 14S 

Double Spiral 141 

Examples of Spiral 140 

Globoid Spiral 142 

" Hoisting 144 

Parallel 133 

Precision 139 

" Single Tooth 165 

" Spiral 13S 

Stepped Bevel 141 

'■ Table of Cast Iron 144 

" Teeth for Hyperboloidal 13S 

Transmission 144 

Gear Teeth, Action of 129 

Construction of Spur.. 128 
Epicycloidal and Evo- 

lute Compared 135 

Evolute Interchange- 131 

able 131 

Friction of Spiral .... 140 

Interchangeable 130 

Internal 131 

Line of Action of 129 

Loss in Determined 

Geometrically 134 

of Circular Arcs. ..... . 131 

Pin 132 

Pressure on 146 

Section of 144 

Stress in. . . . 145 

Thumb Shaped 134 

Wear on 134 

" Tooth Outlines, General Solu- 
tion 129 

" Wheel Arms, Table of 149 

" Hubs 150 

" Plane 136 

" Wheels, Arms of 149 

" " Circumferential Divis- 
ion of 128 

" Classified 127 

" Conical 135 

'' Dimensions of 147 

" Diametral Pitch of... 128 

"' Hyperboloidal 136 

" Interchangeable 128 

" " Pitch of 144 

" Pitch Radius of 12S 

" Rim of 147 

" " Weight of 150 

Gemorsch Chain 1S2 

General Form of Toothed Ratchets. 158 
General Remarks upon Ratchet Me- 
chanism. 171 

Generation of Cycloidal Curves.... 130 

Geneva, Sluice Gates at 275 

Geneva Stop 165 

Gerber's Bolt 57 

Geyser Pump, Siemens' 222 

Gidding's Slide Valve Experiments. 285 

Giffiard's Injector 222 

Girard Turbine 220 

Githen's Rock Drill 231 

Globe Valve 279 

Globoids 142 



Globoid Spiral Gears 142 

Globoid Worm Gearing 143 

Gooch's Link Motion 235 

Goodwin's Split Pulley 194 

Gonzenbach's Expansion Gear 236 

Graham's Escapement 169 

Graphical Addition 26 

' ' Calculation of Axle 86 

" " of Powers ... 24 

" " of Shafting. . . 95 

" " of Crank Axle 106 
" " of Multiple 

Crank Shaft 107 

" " Return Crank 105 

" " Single Crank. 104 

Graphostatics, Elements of 2 2 -38 

Green's Canal Lift 227 

Greindl's Pump 221 

Gresham's Injector _ 222 

Grip Pawl for Cables 185 

Grooves for Rope Transmission .... 195 

Grooved Fly Wheels 195 

Grouped Columns 84 

Group Riveting 41 

Guide Mechanism for Pressure Or- 
gans 217 

Guide Pulleys for Belting 186 

Guides and Guide Bars 121 

Bored 122 

for Marine Engines 122 

Guide Sheaves 185 

Guides, Locomotive 122 

Guiding by Pressure Organs 216 

Guiding, Tension Organs for 172 

Gun Lock Mechanism 166 

Gun Locks 163 

Guns, Hooping of 16 

HAIR SPRING i6g 

Hair Trigger 168 

Half Journals 64 

Hammer, Merrill's Drop 123 

Hand Cranks 109 

Hanger Boxes, Sturtevant's 74 

Hangers 68 

Hangers, Adjustable 74 

Hanger, Sellers' 74 

Hanger, Sellers' Countershaft 74 

Hangers for Rope 181 

Post 73 

Proportion of 73 

Ribbed 73 

Harlow's Valve Gear 231 

Hartwich's Cable Ferry System 175 

Hastie's Steering Gear 236 

Hauling System, Riggenbach's 172 

Haulage Systems, Pennsylvania. . . . 174 

Hawkin's Worm Gearing 143 

Helfenberger's Regulator 236 

Hemp Rope 178 

" Transmission, Specific 

Capacity of 195 

" Wear on 196 

" Weight of 178 

Hero's Fountain < zSS 

Heusinger's Link Motion 235 

Heuberger's Chain Propeller 176 

Hick, Experiments on Stuffing Box 

Friction . . 254 

Hick's Stiffening Ring 269 

High Duty Pumping Engine, Worth- 

ington 232 

Hipp Escapement , 168 

Hirn'S Experiments on Journals.... 64 

Hodgkinson's Experiments 13 

Hofmann's Valve Gear 163 

Hoist, Althaus' 173 

Hoisting Devices 172 

Hoisting Gears 144 

Hollow Columns 83 

Hollow Journals 62 

Hooke's Coupling 97 

Hooks 1 84 

Hooping 45 

" by Shrinkage 45 

of Guns 16 

Hoppe's Accumulator 264 



ALPHABETICAL fXDEX. 



307 



Hose .11 252 

Howaldt's Metallic Packing 254 

Hubs for Rock Arms 102 

Hubs of Gear Wheels 150 

Hunting Valve 238 

Hurdy Gurdy Wheel 220 

Hydraulic Accumulators 218, 264 

Cranes 228 

Elevator 227 

Lever 217 

" Lever, Differential 218 

" Parallel Motion 2r8 

" Power Distribution 219 

" Power Distribution, Ring 

System 256 

" Presses, Cylinders for. . . 243 
" Press, Thickness of Cyl- 
inder 16 

" Hydraulic Ram, Montgol- 

fier's 232 

" Riveting Machine, Twed- 

dell's 228 

" Steering Gear 237 

Tools 228 

'• Transformer 218 

Hyperboloidal Gear Wheels 136 

" Gears, Teeth for 138 

" Wheels, Base Figures 

for,, 136 

IDEAL BENDING MOMENT.... 13 

Ideal Twisting Moment 13 

Impact Water Wheels 220 

Inclined Cable, Transmission with - . 200 

Independent Step Bearings 76 

Inertia, Moment of 5-7 

Influence of Pulley Diameter on 

Wire Rope 197 

Influence of Weight of Wire Rope. 180 

Injector, Giffard's 222 

Injector, Gresham's 222 

Interchangeable Gear Teeth 130 

Interchangeabe Gear Wheels 128 

intermittent Gearing, Brauer's 165 

Internal Flow 240 

" Forces, Equilibrium of .... - 28 

Gear Teeth 131 

Ratchet Wheels 151 

Strap Brakes 215 

Intze's Discussion of Tanks 261 

Inverted Siphon 244 

Iron Columns, Calculations for 82 

Iron, Weight of Round ■ • 55 

Isochronous Escapements 167 

Isolated Forces in One Plane 26 

Isolated Forces, Resultant of 29 

JACKSON'S CLUTCH 101 

Jacquard Loom 163 

James Watt & Co., Thrust Bearing 77 

Jam Nut 56 

Jaw Clutch gS 

Jensen's Worm Gearing 143 

Jet Action 241 

' ' Mechanism 240 

" Propeller 223 

" Pump, Friedmann's 222 

Joints, Expansion 245 

" Flange 58 

Strength of Riveted - 40 

Joint, Universal 97 

Jopling's Water Meter 239 

Journal Boxes, Various forms 69 

" Friction in Rope Transmis- 
sion 195. 205 

'• "of Chain Transmis- 
sion 213 

Journals, Attachments of 67 

" Dimensions - 61 

" Examples of 62 

Fork... 63 

" for Levers loi 

for Shafting 94 

" Friction of 64 

Half 64 

Hollow,,,,, 62 



Journals, Lateral 61 

Lubrication of 61 

" Multiple 63 

" Neck 62 

" Overhung 61 

" Pressure on 61 

" Proportions of 61 

' ' Stress on 61 

" — Various Kinds 60 

KAHLENBERG CABLE ROAD- . 173 

Kaselowsky's Fire Box 269 

Kennedy's Water Meter 239 

Kernaul's Key 49 

Keyed Connections 47 

Keying 47 

" Foundation Bolts 48 

" Peters' Method 102 

" Sci-ew Propellers 49 

Key, Kernaul's 49 

Keys, Concave 48 

" Draft 48 

" Edge 49 

" Flat. . 48 

" for Rock Arms. 102 

Longitudinal 48 

" Methods of Securing 50 

' ' Recessed 48 

Stresses on 48 

' ' Taper of 47 

' ' Unloaded 49 

Kirchweger's Steam Trap 228 

Kirkstall Forge Rolling Mill 126 

Kirkstall Forge, The 94 

Kley's Pumping Engine 233 

Klug's Valve Gear 235 

" Knot " in Cord Polygon 27 

Koechlin's Friction Clutch 100 

Krauss' Connecting Rod End 113 

Krauss' Piston 252 

Kriiger, Investigation of Rivets ... 39 

LACING, BELT 191 

Lagarousse Ratchet 164 

La Louviere Canal Lift 227 

Lamb's Escapement 168 

Langen Gas Engine 161 

Lap Joints, Riveted 40 

Lap of Slide Valve 225 

Latch Bolt 166 

Lateral Bearings 68 

Lateral Journals 61 

Lead Pipe Connections 251 

Lemielle's Ventilator 82 

Lepaute's Escapement 169 

Levasseur's Metallic Tubing 252 

Lever Arms, Calculation of . . . . .... 103 

" Arms of Combined Section. . . 103 

" Differential Hydraulic 218 

Hydraulic 217 

Levers, Combined no 

Equalizing 32 

" Journals for loi 

" Simple loi 

Lifting Frame for Screw Propeller. 151 

Lift Valves 223, 273 

Liquids, Pressure Escapement for 

Moving 22S 

Limit of Elasticity i 

Limit of Elasticity in Beams 8 

Line of Action of Gear Teeth 129 

Line Shafting 93 

Link Couplings gS 

" Motion, Allan's 235 

" " Fink's 235 

" " Gooch's 235 

" " Heusinger's 235 

" " Stephenson's 234 

Load, Breaking i 

Load Length of Wire Rope 180 

Lock, Bramah 166 

" Chubb 166 

•' French 166 

Locking Ratchets 1 50 

Locomotive Axles 107 

" Connecting Rods 116 



Locomotive Connecting Rod, Solid 

End for 113 

" Cross Heads 120 

Guides 122 

" Springs, Screws for. .. . 58 

Locks, Canal 227 

Door 166 

Gun 162 

Lock, Yale 166 

Logarithmic Spiral 26 

Long Distance Fluid Transmission . 233 
Long Distance Power Transmission 259 

Longitudinal Keys 48 

Loom, Jacquard 163 

Loss in Gear Teeth Determined 

Geometrically 135 

Loss in Hemp Rope Transmission. . 195 

Lubrication of Journals 61 

Lucerne, Cable Incline at. , 173 

MACHINE ELEMENTS, CON- 

struction of 39-289 

Machine Riveting 39 

Mackay & McGeorge, Riveting Ma- 
chine Ill 

Magdeburg-Bodenbacher Chain.... 183 

Maltese Cross Gear 165 

Manholes 289 

Mannesmann Tubing 243 

Marine Cross Heads 121 

" Engine Guides 122 

Propulsion 222 

Marshall's Valve Gear 235 

Materials — Strength of 1-21 

Mathematical Tables 291 

Maudslay, Method of Securing Bolts 58 

Maudslay, Thrust Bearing by 77 

Mauser's Revolver 165 

Mazelline's Duplex Pump 231 

Measurement of Fluids, Escape- 
ments for 239 

Measuring Devices, Running 239 

Mechwart's Friction Rollers 127 

Medart Pulley 193 

Merrill's Drop Hammer 123 

Metaline Bearings 179 

Metallic Piston Packing 253 

Metallic Tubing 252 

Metals, Weight of Sheet 43 

Meter for Alcohol, Siemen's 239 

Methods of Securing Bolts 57 

Methods of Securing Pawls 153 

Metrical Screw Systems 52 

Metric Bolts and Nuts 55 

Metric Pipe Thread System 250 

Meyer's Valve Gear 236 

Mill, Bogardus 126 

Minotto's Bevel Friction Wheels... 125 

Mixed Tooth Outlines 133 

Mines, Chain Transmission in 213 

Modulus of Elasticity i, 13 

" Resistance.... i 

Rupture 1,2 

Transmission 20S 

Molinos & Pronnier, Speed of Rivet- 
ing 39 

Moment of Inertia 3, 5, 7 

Moment of Inertia, Polar 11 

Mont Cenis Air Compressors 232 

Montejus 22S 

Montgolfier's Hydraulic Ram 232 

Morin's Experiments on Journals. . . 64 

Motors for Pressure Organs 219 

Moulding 240 

Moxon's Belt Fastening 193 

Mudge's Escapement 168 

Muff Coupling 95 

Mule Post 188 

Mule, Spinning 169 

Multiple Collar Bearings 77 

" Collar Thrust Bearings. .. . 66 

" Crankshafts 107 

Journals 63 

Ratchets 154 

Supports for Bearings 80 

' ' Trussed Beams , , , , 35 



;o8 



ALPHABETICAL INDEX. 



Multiple Valves 276 

Mviltiplication and Division Com- 
bined 23 

Multiplication by Lines 22 

Murdock's Slide Valve 234 

Muschenbroeck's Pump 223 

NAGEL TURBINE 220 

Napier's Clutch 101 

Napier's Differential Brake 214 

Natural Reservoirs 21S 

Neck Journals 62 

Neck Journals, Connections for. ... 114 

Negative Reservoirs 219 

Neutral Axis 3 

Neutral Plane 10 

Neustadt's Chain 1S3 

Newcoraen Engine 163 

Normandy's Pipe Joint 249 

Norton's Pump 225 

Nut, Jam ■ 59 

Nut Locks 56 

Nuts, Washers and Bolt Heads 54 

OBELISK, FORCES IN RAISING 2S 
Oberursel, Rope Transmission at. 203, 205 
Oeking's Water Counter-balance... 217 

Oil Tanks 21S 

Oldham's Coupling 96 

Open Belts, Cone Pulleys for 189 

Open Link Chain 182 

Ordvvay, Experiments on Pipe Cov- 
ering 245 

Oscillating Fly Wheel 233 

Oscillating Pumps 226 

Oscillation of Escapements, Time of 167 

Osterkamp's Rope Hanger 181 

Otis Elevator 228 

Overhead Cable Tramv^ay 175 

Overhung Journ als 61 

PACKING FOR HYDRAULIC 

Press 253 

" for Pump Pistons 255 

" Howaldt's Metallic 254 

*' Piston 216 

" Standard Prussian Rail- 
way . 255 

Pagel's Elastic Washer 57 

Pallets 168 

Pappenheim Chamber Wheel Train . 219 

Parabola, Biquadratic 10 

Parallel Forces — Equilibrium of. .30, 31 

Gears 133 

" Motion, Hydraulic 218 

Rods for Locomotive En- 
gines 117 

Parson's Bolts 57 

Pattison's Pump 226 

Pawl, Cable Grip 185 

Pawl. Saladin's Friction ... 161 

Pawls for Chains - 185 

" Friction 159 

Methods of Securing 153 

Pawl, Spring 153 

Pawls, Release of Friction 161 

Pawl, Thrust upon 152 

Pawl, Thumb Shaped 160 

Payton's Water Meter. 220 

Pedestal Bearings 71 

Penn's Connecting Rod End 113 

Method of Securing Bolts.. 57 

Piston 252 

Pennsylvania Haulage Systems.... 174 

Periodical Escapements 169 

Periodical Pressure Escapements... 229 

Peters' Method of Keying 102 

Petit's Pipe Joint 248 

Pfalz-Saarbruck Screw Thread Sys- 
tem 53 

Phoenix Column 59. 83 

Physical Ratchet Tram 171 

Pickering's Steam Pump 230 

Pillow Blocks 68 

Blocks, Adjustable 69, 70 

Block, Sellers' 70 



Pillow Blocks, Large 69 

Blocks, Proportional Scale for 68 

Block, Sturtevant's 70 

Pin Gearing, Double 132 

" Gear Teeth 132 

■' Ratchet Wheel 152 

Pins, Crank 61 

Pin, Split 56 

Pipe Connections, Socket 248 

' ' Coverings 243 

' ' Fittings 249 

" Joint, Normandy's 249 

" Petit's 248 

Riedler's 249 

' ' Riveted 244 

Pipes, Ball Joints for 249 

Pipes, Connections for Cast Iron . . . 248 
" Connections for Wrought Iron 249 

Copper 246 

Flange Joints for 248 

Flexible 252 

for High Pressures 242 

Resistance of Bends in 247 

Resistance to Flow in 246 

Pipe Sockets 250 

Pipes, Steam 245 

Pipe, Steel 243 

Pipes, Thickness of Cast Iron 242 

Pipe Threads, Briggs' System 250 

j" Thread System, Metric 250 

" Weight of Cast Iron 242 

" Wrought Iron 243 

Piston, Krauss' 252 

Packing 216, 253 

Packing, Metallic 253 

Penn's 252 

Pumps 223 

Rods 255 

Pistons 216, 252 

Piston, Swedish 253 

Pistons with Valves 286 

Piston Valves 286 

Pitch and Face of Gearmg, Calcula- 
tion of 144 

Hoisting Gears. , 144 
Transmission 

Gears . 144 

Chain 183 

Circles, Table of 1-28 

" of Gear Wheels 144 

" Radius of Gear Wheels 128 

Pivot Bearings, Flat 66 

Pivots, Formula for 65 

" Pressure on 65 

Proportions of 65 

Plain Slide Valve 282 

Plain Slide Valve Gear 234 

Plane Gear Wheel 136 

Planing Machine, Ducommun & Du- 

bied's 176 

" " Sellers 176 

" Shanks' 163 

Plate Coupling 95 

Plungers 216, 253 

Plunger Pumps 223 

Pneumatic Power Distribution 257 

Pneumatic Tube 227 

Polar Moment of Inertia 11 

Polishing Belts 177 

Polonceau's Connecting Rod End.. 114 

Polygons, Area of 24 

Poncelet's Chain 173 

Poncelet's Water Wheel 220 

Porter- Allen Connecting Rod 117 

Porter- Allen Engine 236 

Post Hangers 73 

Pouyer's Coupling loi, 152, 153 

Powel Valve Gear 163 

Power Distribution, Compressed Air 219 

" " Hydraulic 219 

" " Hydraulic Ring 

System 256 

" " Pneumatic 257 

" " Steam 219 

" " Systems 219 

'■ " Vacuum 219 



Power Escapements 169 

Powers, Graphical Calculation of , . . 24 
Powers of Trigonometrical Func- 
tions 25 

Power Transmission by Superheated 

Water 219 

Practical Resistance. i 

Precision Gears 139 

Precision Ratchets 157 

Prentiss' Coupling 216 

Pressure Escapements for Moving 

Liquids 228 

on Gear Teeth 146 

Journals 61 

'■ Lift Valves 277 

" Pivots 65 

Screw Threads 58 

Organs , 216 

" Conductors for 242 

" " Conduits for 216 

" " Escapements for... 226 

" " Guidmg by 216 

" Guide Mechanism 

for . 217 

" " Motors for 219 

" " Ratchet Mechanism 

for 223 

" Reservoirs for. .218, 260 
" ■' Running Mechanism 

for 219 

" '■ Technological Appli- 
cations of 250 

" Superficial i 

Transmission, Specific Ca- 
pacity of 255 

Proof Diagrams of Axles 87 

Propeller Bearing, Ravenhill & 

Hodgson's . 74 

Jet 223 

Lifting Frame for 151 

' ' Screw 223 

" Shafts, Couplings for 95 

Propelling Chain : 1S5 

Proportions of Axles 85 

" Chain. 183 

" Flange Joints 59 

" Hooks 184 

" Journals 61 

Pivots 65 

" Pulleys 193 

Propulsion, Marine 222 

Prussian Standard Car Bearing 75 

Pulley Block, Weston's Differential. 174 

" by Walker Mfg. Co 193 

" Diameter, Influence on Wire 

Rope 197 

Fowler's Clamp 203 

Medart 193 

Pulleys, Balancing of 194 

" Cone 189 

Construction of Rope 202 

'• Crown of Face 186 

"' Double Arm 193 

Guide 186 

" Fast and Loose 188 

for Cable, Arrangement of . 202 

Proportions of 193 

Split 193 

Pulley Stations, Construction of . . 204 

Pulleys, Tightening 186 

Pulley, Sturtevant's 194 

Pulleys, Umbrella 207 

Pulleys, Vertical Supporting 188 

Pulsometer 232 

Pump, Althaus' 223 

Amos & Smyth's 224 

Bag 217 

Bellows 217 

" Diaphragm 217 

Donnadieu's 223 

Downton's 224 

Friedmann's Jet 222 

Gears, Adjustable 236 

" Greindl's 221 

" Engine, Kley's 223 



ALPHABETICAL IXDEX. 



309 



Pumping Engine, Worthington's 

High Duty 232 

Pumping Machinery 229 

Pump, Mazelline's Duplex 231 

Muschenbroeck's 233 

" Norton's 225 

Pattison's 226 

Pistons 253 

Pistons, Packing for 255 

Regulator, Helfenberger's . 236 

Repsold's 221 

" Rittinger's 223 

Pumps, Centrifugal 222 

" Considered as Ratchet 

Trains 223 

" Double Acting 224 

Pump, Siemen's Geyser 222 

Pumps, Oscillating 226 

Pump, Spiral ■■■ ■ 221 

Pumps, Piston 223 

Plunger 223 

Rotary 226 

Stolz's 224 

Pump, Stone's 224 

" Valve Gear 225 

Valves, Riedler's 278 

Vose's 224 

Worthington's Duplex 231 

QUADRANTS 153 

Quadrilateral Figures, Area of 25 

Quarter Twist Belts. . ■ • 186 

Quarter Twist Belt, Shifter for 189 

RACK, RATCHET 151 

Rack Teeth, Evolute 132 

Railway Axles 88 

Ramsbottom's Crane 176 

Ramsbottom's Friction Clutch gg 

Ratchet, Anchor 155 

Brace, Weston's 154 

Clamp 160 

" Crown 154 

Cylinder 156 

" Dobo's 160 

Gearing 150 

Cylinder 156 

" " Dimensions of Parts 

of 158 

Gears, Toothed Running. . 150 

Lagarousse 1 64 

Mechanism for Pressure 

Organs 223 

General Re- 
marks upon 171 
'" " Kinematically 

Discussed.. 171 

Rack 151 

Rod Friction 163 

Ratchets, Checking ....... 150 

Continuous 150 

" Continuous Running 164 

" Double Friction 160 

Friction 158 

" General Form of Toothed 158 

" Locking 150, 166 

" Multiple 154 

" of Precision 157 

■' Releasing 150, 162 

" Running 1 50 

" Running Friction 158 

" Silent 153 

Spring 153 

" Stationary 1 50, 156 

" Stationary Friction 161 

Step 155 

Step Anchor 157 

Throttle 161 

" with Locking Teeth, Con- 
tinuous 165 

" Teeth, Flanks of 153 

" Teeth, Form of 150 

•' Tension Organs 185 

" Tooth, Dead . 152 

" Train, Physical 171 



Ratchets, Trains, Chemical 171 

Wheels, Internal 151 

Wheels, Special Forms... 154 

'• Wilber's 153 

Ravenhill & Hodgson, Propeller 

Bearing 74 

Ravenhill & Hodgson, Thrust Bear- 
ing by ; 77 

Reciprocating Valve Gears 234 

Regulator, Guhrauer & Wagner's . . 237 

Regulator, Rigg's 236 

Reichenbach's Water Pressure En- 
gine 22g 

Releasing Ratchets .150, 162 

Releasing Valve Gears 162 

Release of Friction Pawls 161 

Rennie, Experiments on Journals. . . 64 

Repeating Watches 169 

Repsold's Chamber Gear Train 220 

Repsold's Pump 221 

Reservoirs for Air and Gas 272 

" for Gases 219 

" for Pressure Organs. 218, 260 

Natural 218 

" Negative 219 

Resistance, Coefficients of 

" Modulus of 

" of Bends in Pipes 24' 

" of Valves in Pipes 24' 

Practical 

Theoretical . . : 

" to Bending 2 

" Buckling 13 

Flow in Pipes 246 

" Shearing 2 

Torsion 11 

Resultant of Isolated Forces 29 

Load on Water Wheel 

determ'd Graphically. 34 

" Several Forces 26 

Return Crank 105 

Return Crank, Graphostatic Calcu- 
lation for 105 

Reuleaux's Coupling g6 

Escapement 168 

" Friction Clutch 100 

" System of Rope Trans- 
mission 206 

" Valve Diagram 234 

" Winding Drum 173 

Reversing Gear, Globoid 143 

Revolver, Mauser 165, 166 

Rhenish Railway Cable 174 

Ribbed Axles 91 

Rib Profiles, Construction of gi 

Richard's Manometer 288 

Rider's Valve Gear 236 

Riedler's Air Compressors 27g 

Pipe Joint 249 

Valve Gear 278 

Riggenbach's Cable System 174 

Riggenbach's Hauling System 172 

Rigg's Regulator 236 

Rigid Couplings 95 

Rim of Gear Wheel 149 

Ring System of Cable Transmis- 
sion 208-211 

Rittinger's Pump 223 

Riveted Joints, Construction of An- 
gles 44 

" " Junction of Plates.. 43 
" " Proportional Scale 

for 41 

" ■' Reinforcement of 

Plates 44 

" " Special Forms 43 

Strength of 40 

Table of 40 

Pipe 244 

Pipes, Flanges for 249 

Rivet Heads, Proportions of 39 

Riveting 39 

Boiler 42 

" Group. 41 

Machine 39 

" " Mackay & McGeorge in 



Rivet'ng Machine, Tweddell's Hy- 
draulic 228 

Riveting, Speed of 39 

Rivets 39 

Robinson's Experiments on Lift 



Valves. 



277 

Robertson's Friction Wheels 125 

Rock Arms 162 

Rock Drill, Githen's 231 

Rod Connection, Wiedenbruck's. ... 50 

Rod, Friction Ratchet 163 

Rods, Connecting 112 

Rolled Shafting 94 

Roller Bearing, Cambon's 127 

Bearings 126 

" Bearings for Sheaves 179 

Roof Trusses, Force Plans for 36 

" Truss, Polygonal 37 

" with Simple Principals 36 

" with Trussed Principals 36 

Root's Blower '.x 221 

Roots, E.xtraction of . 26 

Rope, Centrifugal Force of Wire. . . . 197 

Connections 181 

Cotton .^. i7g 

Cross Section of Wire........ 196 

Curve, Construction of 202 

Hanger, Osterkamp's 181 

Hangers 181 

Hemp 178 

Influence of Pulley Diameter 

on 197 

Ropes of Organic Fibres 178 

Rope Pulleys, Construction of 202 

Ropes, Creep of. 202 

" Deflection of Wire 198 

Flat 178, 181 

' ' Loss from Stiffness 196 

Rope, Specific Capacity of Wire. . . . 196 

Rope Splice 181 

Ropes, Stiffness of 181 

" Tightened Driving 200 

Ziegler's Experiments on... 181 

Rope Transmission 194 

at Bellegarde. . . . 205 

" at Freiburg 205 

" at St. Petersburg 205 

" at Schaffhausen. 204 

" at Zurich 205 

" Cotton 196 

Cross Section for 

Hemp 195 

" Efficiency of.... 205 
Loss in Hemp. . . 195 
" Reuleaux's Sys- 
tem of 206 

" Specific Capacity 

of Hemp 195 

Wire ig6 

Weight of Hemp 178 

Wire 179 

Rotary Pumps 226 

Valves 281 

Valve, Wilson's 285 

Rotative Motors, Adjustable Gears 

for 237 

" Pressure Engines 233 

Valve Gears 234 

Round Connecting Rods ij6 

Round Valves 275 

Roux's Water Pressure Engine 229 

Rubber Springs 21 

Rubber Springs, Werder's Experi- 
ments 21 

Running Chains 182 

" Friction Ratchets 158,160 

" Mechanism for Lifting 

Water 221 

" Mechanism for Pressure 

Organs 219 

" Ratchet Gears, Toothed. . 150 

" Ratchets 150 

" Ratchet Trains, Fluid.... 223 

" Tension Organs 172 

Rupp's Variable Speed Gear 124 

Rupture, Modulus of i 



;io 



ALPHABETICAL INDEX. 



SAFETY, COEFFICIENT OF.. .. i 

Safety Devices for Elevators 164 

Safety, Factor of i 

Sail Boat 223 

St. Petersburg, Rope Transmission at 205 

St. Louis Bridge 60 

Saint Venant, Friction of Water.... 247 

Saladin'.s Friction Pawl 161 

Sanderson's Gas Meter 239 

San Francisco Cable Tramways. . . 174 
Saxby & Farmer, Signal Apparatus. 166 

Saw, Zervas' Wire 177 

Saws, Band • i77 

Scale Beams m 

SchafEhausen, Rope Transmission at 204 

Schiele Turbine 220 

Schmick's Canal Cable System 175 

Schmid's Water Meter 240 

Schmid's Water Pressure Engine . 236 

Schurman's Clutch loi 

Schurman's Friction Coupling 215 

Screw, Archimedian 221 

" Connections 5S 

" Propeller 223 

Propeller, Lifting Frame for 151 

". Propellers, Built up 57 

" Propellers, Method of Keying 49 

Screws, Enlarged 58 

Screw Thread, Construction of 50 

" " Dimensions of V. .. . 50 

" " Friction of 51 

" " Pressure on 58 

" Threads, Special Forms of . .. 57 

" Threads, Trapezoidal 58 

Section of Gear Teeth 144 

Section Modulus 5. 7> n 

Sections of Uniform Resistance.... 8 

Secured Bolts . , 57 

Securing Keys, Methods of 5° 

Segner s Water Wheel 220 

Self Guiding Belting 186 

Seller's CoupUng 96 

" Friction Feed 126 

' ' Hanger 74 

Pillow Block 70 

'■ Planing Machine 176 

Screw Thread System 52 

Wall Bearing 71 

Sewage System of Berlin 219 

Sewing Machine Check 151 

Shafting .,..■■ 92 

Deflection of • 94 

" Dimensions of 92 

■" Examples of Torsion in 94 

" Graphical Calculation of - . 94 

Journals for 94 

Line 93 

Rolled 94 

" Specific Capacity of 257 

" Torsional Deflection of 92 

" Wooden 94 

" Wrought Iron 93 

Shank's Planing Machine 163 

Sharp's Coupling 96 

Sharp's Strap End 112 

Shearing, Resistance to 2, 10 

Shearing Strain 2 

Sheaves, Chain 185, 211 

Sheaves, Roller Bearings for 179 

Shield Gearing 133 

Shifter for Quarter Twist Belt 189 

Shifters, Belt.... • . 188 

Shifting Eccentrics 235 

Short Span Cable Transmissions. . . . 200 

Shrinkage, Hooping by 45 

Shrinking Fit 17 

Fits, made with Boiling 

Water 47 

" Rings, Clerk's Method. . . 45 

" Temperatures 45 

Sickles' Adjustable Valve Gear 237 

Sickles' Valve Gear 162 

Side Wheel Steam Boat 223 

Siemens & Halske, Electric Signals. 166 

Siemens' Alcohol Meter 239 

Siemens' Geyser Pump 222 



Signal Apparatus, Saxby & Farmer. 166 

Silent Ratchets 153 

Simple Crank Axle 106 

Simple Escapements 167 

Single Acting Steam Engine 229 

Single Tooth Gears 165 

Sinoide gi 

Sinoide, Cj'cloidal 13 

Siphon, Direct 287 

Siphon, Inverted ... .244', 287 

Slide Valve, Common 225 

" " Gear, Plain 234 

" " Lap of 225 

-" Murdock's 234 

" Valves. . . 223, 273, 2S1, 282 

" Valves, Balanced 285 

Sliding Brakes 215 

Sliding Crank 226 

Slipper Cross Head 121 

Sluice Gates at Geneva 275 

Sluice Valve 2S1 

Snail 169 

Solid End for Connecting Rod 113 

Special Forms of Bearings 74 

"of Bolts 55 

" " of Ratchet Wheels. - 154 

" of Screw Threads. .. 58 

Specific Capacity of Belting 190 

of Driving Chains. 211 
" " of Hemp Rope 

Transmission.. . 195 
" " of Pressure Trans- 
missions 255 

" " of Shafting 257 

" " of Wire Rope 196 

Speed Gear, Variable 124 

Spencer & Inglis Valve Gear 262 

Spherical Cycloid 135 

" Journal, Connection for. . 115 

" Spiral 142 

Valves 275 

Spiral Bevel Gears 141 

Gears 138 

" " Double, 141 

" " Examples of 140 

" Teeth, Friction of 140 

Gears, Globoid -. 142 

Pump 221 

" Spherical 142 

' ' Winding Drums 181 

'■ Wire Pipe 252 

Spinning Miile 169, 196 

Splice for Ropes 181 

Split Pin 56 

" Pulley, Goodwin's 194 

" Pulleys 193 

Spring, Dudley's 20 

" Pawl 153 

" Ratchets 153 

Springs, Best Material for 20 

" Calculation of 18 

Table of 18-19 

" Vulcanized Rubber 21 

Spur Gear Teeth, Construction of. . 128 

Squaring Device for Cranes 172 

Square Thread 40 

Standing Tension Organs 172 

Stand Pipes 2S7 

Statical Moment 3 

Statical Moment, Graphically Con- 
sidered 33 

Starting Valve 281 

Star Pin 153 

Stationary Chains 182 

" Friction Ratchets 161 

" Machine Elements 2S9 

" Ratchets 150, 156 

" Valves 289 

Steam Boat, Side Wheel 223 

' ' Boilers 265 

" Distribution of Power 257 

Engine, Single Acting Steam 229 

" Pipes 245 

" Power Distribution 219 

" Pump, Blakes 240 

" " Deane's 230 



.Steam Pump, Pickering's 230 

Tangye's 230 

Steering Gear 238 

Trap, Kirchweger's 228 

Steel Pipe 243 

Steering Gear 171 

" Britton's 238 

" " Davis & Co.'s 238 

" Douglas & Coulson's. 238 
" " Dunning & Bossiere's 238 

" Hastie's 238 

" " Hydraulic 237 

" Steam 238 

Steib's Ventilator 222 

Step Anchor Ratchet 156 

" Bearings '75 

" Bearing, Support for , 80 

" Bearings, Wall. . . . ^ 75 

" Gearing 141 

Stephenson's Link Motion 235 

Stepped Bevel Gears 141 

Step Ratchets 155 

Step Valves 276 

Stcjvart, Experiments on Springs... 21 

Stiffness of Belts 194 

of Ropes 181 

'* " Eytelwein's For- 
mula 181 

" " Loss from 196 

" " Weisbach's For- 
mula 181 

" •' Wire Rope 206 

Stolz's Pump 224 

Stone's Pump 224 

Stop, Geneva 165 

Storage Reservoirs, General 273 

Strain, Shearing 2 

Strains of Flexure ... 3 

Strap Brakes 211, 215 

Brakes, Internal 215 

End for Connecting Rod 112 

" End, Sharp's 112 

Straps, Eccentric 115 

Strength of Cast Iron Columns 83 

of Materials 1-2 1 

of Wire Rope 179 

" Tensile i 

Stress Curve 87 

Stresses, Compound 13 

" in Columns , 82 

' ' on Keys 48 

Stress on Belting 191 

on Gear Teeth 145 

' ' on Journals 61 

" S, Value of 8 

Striking Mechanism for Clocks 169 

Stub End, for Fork Journal 114 

Stuffing Boxes 253 

Box, Farcot's 254 

" Box, Friction in 254 

Sturtevant's Hanger Boxes 74 

Pillow Block 70 

Pulley 194 

Superficial Pressure • i 

Superheated Water Transmission , . . 219 

Supporting Pulleys, Vertical 188 

Supports for Bearings 79 

" " General Prin- 
ciples 82 

" Simple 79 

Supporting Power of Beams 5 

Swedish Piston .... 253 

Swedish Railway, Boilers for 272 

Sweet's Valve Gear 235 

Swivels 182 

Swivels for Chain 184 

Symmetrical Simple Axles S3 

TABLE OF BEAM SECTIONS. 5, 6, 7 
Tables of Curves, Areas and Vol- 
umes 291-296 

Table of Numbers 300-301 

Tackle Block 172 

Tangential Pressure on Crank Pin . . 233 

Tangye's Crane 176 

Tangye's Steam Pump 230 



ALPHABETICAL INDEX. 



311 



Tanks, Cast Iron 260 

" Combination Forms for 264 

" Intze's Discussion of 260-264 

" Oil 21S 

" With Concave Bottoms 262 

" Wrought Iron 260 

Taper of Keys 47 

Technological Applications of Pres- 
sure Organs 240 

Technological Applications of Ten- 
sion Organs i77 

Tenacity i 

Tensile Strength 1 

Tension Organs 172 

" for Driving 173 

" " for Guiding 172 

" " for Winding 172 

" Ratchet 185 

" " Running 172 

" " Technological Appli- 
cations of 177 

' ' Resistance to 2 

Tests for Chain . 1S3 

T Fittings 251 

Theoretical Resistance i 

Thickness of Cast Iron Pipes 242 

Thick Vessels, Walls of 16 

Thomas' Calculating Machine. . . 153, 146 

Thometzek's Valve 276 

Thomson's Turbine. 220 

Three-part Bearings 70 

Throttle Valves 161 

Throttle Vaves 279 

Thrust Bearing by James Watt & Co. 77 

" by Maudslay 77 

" by Penn 77 

" " by Ravenhill & 

Hodgson 77 

" " Compound Link as 67 

" Bearings 65, 68, 75 

Collar 66 

" " Examples of 78 

Multiple Collar 66 

'■ " with Wooden Sur- 
face* 76 

" upon the Pawl 152 

Thumb Shaped Pawl 160 

Thumb Shaped Teeth 134 

Tiede's Escapement 168 

Tightened Cables, Table for 200 

Tightened Driving Ropes 200 

Tightening Pulleys 186 

Tightening Pulley, Weaver's 186 

Toggle Friction Brake 162 

Tools, Hydraulic 218 

Toothed Gearing 127 

Tooth Friction in Spur Gearing. . . . 134 
" Outlines, General Solution of 129 

" " Mixed . 133 

of Circular Arcs... 131 

Torpedo, Fish 171 

Torsional Deflection of Shafting... 92 
Torsion, Determination of Angle of 93 

Resistance to 11 

Table 12 

'• Uniform Resistance to. .. . 13 

Towne Crane 176 

Transformer, Hydraulic 218 

Tranforming Arm Sections, Table for 103 
Transmission at Long Distance, 

Fluid 233 

Chain 211 

Gears 144 

" Long Distance Power. 259 

" Modulus of 208 

Rope 1 94 

With Inclined Cable. . . 200 
Transportation, Fluid Escapement 

ment for 227 

Transporting Belts 221 

Trapezoidal Screw Threads 51, 58 

Trap, Water 287 

Triangles, Area of 23 

Trick's Valvs 2S4 

Trigger, Hair 153 

Trigonometrical Formulee 299 



Trgionometrical Functions, Powers 

of 25 

" Table ... .297-299 

Trussed Beams, Double 35 

" " Simple 35 

Triple 35 

Tubing, Levasseur's Metallic 252 

Tumbling Gears 163 

Turbine, Borda's : . 220 

" Cadiat 220 

" Fourneyron 220 

Francis 220 

' ' Girard 220 

" Nagel 220 

Schiele : . 220 

" Thomson's 220 

Tweddell's Accumulator 265 

Tweddell's Hydraulic Riveter 228 

Twin Link 184 

Twisting Moments, Graphically Con- 
sidered 33 

Tympanon of Archimedes 221 

UHLHORN'S COUPLING. . . .101, 153 

Umbrella Pulley 207 

Uniform Escapements 167 

Uniformly Distributed Forces 32 

Uniform Resistance, Columns of . . . 13 

" " Sections of . . . . 8 

to Bending. ... 8 

" " to Torsion. ... 13 

Strength, Bodies of 2 

Universal Gears, Beylich's. 136 

Universal Joint 97 

Unloaded Bolt Connections 60 

Unloaded Keys 49 

Unperiodic Power Escapements for 

Pressure organs 227 

VALVE, ALLAN'S DOUBLE.... 2S3 
Armstrong's Supported.. 286 

Bell ■ .. 276 

Boulton & Watt's Bal- 
anced 285 

" Brandau's Double Seated. 286 

" Cave's Balanced 285 

" Corliss 236 

" Cornish 280 

Cramer's Balanced 2S0 

Cuvelier's Underpressure 2S6 

" Double Beat 280 

" D 283 

" " Flap. 274 

" Diagram, Reuleaux's .... . 234 

" Zeuner's. 234 

" Gear, Angstrom's 235 

Brown's 235 

•■ Call & Co 162 

Cam 236 

" " Corliss 162 

" " Cornish 163 

" " for Pumps 225 

" Harlow's 231 

" " Hofmann's 163 

" Klug's 235 

" " Marshall's 235 

" " Plain Slide 234 

Powel 163 

" " Rider's 236 

" Gears for Rotative En- 
gines 234 

" " Reciprocating 234 

" " Releasing 162 

Rotative 234 

" Gear, Sweet's 235 

Gear, Wannich 162 

Globe 279 

" Gridiron 283 

Hick's Double 283 

Injector 279 

Kirchweger's Balanced .. . 2S5 

Lindner's Balanced 285 

Plain Slide 225, 282 

Porter- Allen 2S7 

Rubber Disk 274 

Valves 279 



Valves, Balanced , 279 

Valves, Balanced Slide 285 

Valve, Schaltenbrand's Double Seat- 
ed. 
Valves, 



286 

Check 274 

Closing Pressure of 278 

Conical 275 

Considered as Pawls. . . .223, 273 

Flap 274 

Flat Disk 275 

Fluid 287 

Gate 2S2 

Gidding's Experiments on . . 285 

Lift ... 223, 273 

Mechanically Actuated 27S 

Multiple 276 

Piston 286 

Resistance of 247 

Robinson's Experiments on. 277 

Rotary 2S1 

Round 275 

Slide 225, 273, 281 

Spherical 275 

Valve, Starting 282 

Valves, Stationary 289 

Step 276 

Throttle 279 

Unbalanced Pressure on Lift 277 

Valve, Sweet's Balanced 287 

Valves, Width of Seat 274 

Valve, Thometzek's 276 

Trick's 274 

" Wilson's Balanced 287 

" Wilson's Rotary 286 

Value of Stress S 8 

Vacuum Power Distribution 219 

Variable Speed Gear 124 

Variable Speed Gear, Rupp's 124 

Velocity Curves 233 

Ventilator, Fabry's 221 

Lemielle's 82 

" Steib's 222 

Verge Escapement 168 

Volume, Escapements for Measure- 
ment of 239 

Von Gerike's Air Pump 225 

Vose's Pump 224 

V Screw Thread 50 



WALKER MFG. CO., PULLEY 

by 

Walking Beams 

Wall Bearings 68, 

" Bearing, Support for 

' ■ Bearing, Sellers' 

' ' Brackets 

Walls of Vessels, Resistance of 

Wall Step Bearings 

Wannich Valve Gear 

Washers 

Watches, Repeating 

Water Counterbalance, Oeking's . . . 

" Meter, Jopling's 

" " Kennedy's 

" " Payton's 

" " Schmid's 

■ ' Pressure Engine, Belidor's . . 
" " " Re ic hen- 

bach's. . 
" " " Roux's. . . . 

" " " Schmid's.. 

" Reservoir, of Frankfurt on 

Main 

Rod Connection ■. . . 

Running Mechanism for Lift- 
ing 

" Trap 

" Trap, Morrison, Ingram & Co. 

Wheel, Poncelet's 

" Wheel, Resultant of Load on 

Wheels. Axles for 

Wheel, Segner's 

Wheels, Gravity 

" Wheels, Impact. 

Watt's Condenser 

Wear on Gear Teeth 



194 
no 
71 
79 
71 
72 
15 
75 
162 

54 
169 
217 
239 
239 
220 
240 
229 

229 
229 

236 

218 
233 



287 
288 
220 
34 
91 
220 
2ig 
220 
230 
134 



312 



ALPHABETICAL INDEX. 



Wear on Hemp Rope i . • 196 

Weaver's Tightening Pulley 1S6 

Wedge Friction Wheels 125, 160 

Weighing Machine, Emery's I73 

Weight of Cast Iron Pipe 242 

of Chain 183 

of Gear Wheels 150 

" of Hemp Rope 17S 

" of Round Iron 55 

Sheet Metal 43 

of Wire Rope 180 

Weir, Camere's 275 

Weisbach, Formula for Friction of 

Water 246 

Weisbach's Formula for Stiffness of 

Ropes ■ 181 

Werder, Experiments on Springs. . . 21 
Weston's Differential Pulley Block. 173 

Friction Clutch loi 

Ratchet Brace I54 

Wet Gas Meter 239 

Wheels, Classification of 122 

Whip Action of Connecting Rod. . . 116 
Whitehead Torpedo 237 



Whitworth's Screw System 51 

Whitworth's Pipe Thread Scale.... 51 

Wiedenbruck's Rod Connection.... 50 

Wilber's Ratchet 153 

Wilson's Rotary Valve 286 

Wilson's Water Gas Furnace 288 

Winding Drum, Reuleaux's 173 

" Drums, Spiral 181 

■" Tension Organs for 172 

Windlass 172 

" Brown's 173 

Differential 173 

Windmills 220 

Wind Stresses, Graphically Deter- 
mined 37 

Wire Rope 179 

'• Influence of Weight. .. . 180 

" Load Length of 180 

" " Strength of 1 79 

" Transmission 196 

" Weight of iSc 

" Saw, Zervas' 177 

Wooden Axles, Proportions of 92 

Wooden Shafting • 94 



Worm and Worm Wheel 135 

Gearing, Globoid 143 

Hawkins 1^3 

Jensen's 143 

Worthington High Duty Pumping 

Engine 232 

Worthington's Duplex Pump 231 

Worthington's Equalizer 232 

Wrapping Connections 173 

Wrenches 56 

Wrought Iron Cranks, Single 104 

Pipe 243 

" " Shafting 93 

Walking Beams iii 

YALE LOCK 167 

Yoke Bearings 72 

ZERVAS' WIRE SAW 177 

Zeuner's Valve Diagram 234 

Zimmermann's Belt Shifter . . . 189 

Zuppinger's Water Wheel 219 

Zurich, Rope Transmission at 205 



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